Vector space and 3D flow field

Leo Liu
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Homework Statement
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Relevant Equations
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1626483157058.png

Could someone explain the green highlight to me, please?
 
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The sum emits light of constant speed c with constant power. The emitted photons becomes more sparse at distant places as ##r^{-2}## so the sun is observed more dimmer.
 
anuttarasammyak said:
The sum emits light of constant speed c with constant power. The emitted photons becomes more sparse at distant places as ##r^{-2}## so the sun is observed more dimmer.
Thanks for providing some physical intuition. But is it because the surface density of a mass flow is
$$\frac{\dot m}{A}=\frac{\dot m}{4\pi r^2}$$
?
 
Yes. What's wrong with it ?
 
anuttarasammyak said:
Yes. What's wrong with it ?
Nothing. Just making sure I understand where this proportionality comes from.
 
Another perspective: imagine two concentric spherical surfaces of radii ##\rho_1## and ##\rho_2## (##\rho_2 > \rho_1##) which bound a region ##R##. In steady state the mass contained in ##R## is constant, so the mass fluxes into the inner surface and out of the outer surface are equal: ##4\pi {\rho_1}^2 \delta_1 v = 4\pi {\rho_2}^2 \delta_2 v \, \implies \, \delta \rho^2 = \mathrm{constant}##.
 
Thread 'Use greedy vertex coloring algorithm to prove the upper bound of χ'
Hi! I am struggling with the exercise I mentioned under "Homework statement". The exercise is about a specific "greedy vertex coloring algorithm". One definition (which matches what my book uses) can be found here: https://people.cs.uchicago.edu/~laci/HANDOUTS/greedycoloring.pdf Here is also a screenshot of the relevant parts of the linked PDF, i.e. the def. of the algorithm: Sadly I don't have much to show as far as a solution attempt goes, as I am stuck on how to proceed. I thought...
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