When we talk about the Robertson-Walker scale factor in the early universe or the matter inside a black hole event horizon, there must be an arbitrarily large or infinite density.   But how can this be possible if we consider Fermi-Dirac statistics? I mean suppose the scale factor decreases toward early times. Can we squeeze the matter as much as the degeneracy pressure allows us in order to put a large amount of matter inside a limited physical space? How can we imagine a typical galaxy inside a smaller space that is not expanded as much as today, let alone all the galaxies in the universe? The same is true about a black hole. How can we explain infinite density and growing matter inside a limited physical space? Even if we say that their momentum quantum states are respecting the statistics but again there are particles which occupy physical space, which is by assumption far smaller than the current epoch or outside of a black hole event horizon.
A star's collapse can be halted by the  due to the Pauli exclusion principle. In extreme relativistic conditions, a star will continue to collapse regardless of the degeneracy pressure to form a black hole. Does this violate the Pauli exclusion principle? If so, are theorists ok with that? And if it doesn't violate the Pauli exclusion principle, why not?
Command for argmin or argmax? How can you get the argmin or argmax in Latex? Two solutions I found are:  \underset{x}{\operatorname{argmax}}   \DeclareMathOperator*{\argmin}{arg\,min}   Any other ideas?  [1]   [2] Change the location of saved screenshot when screenshot taken by printscreen button in GNOME 3 I want to change the default location of "Save a screenshot to Pictures" to save it to Desktop.   If that is not possible, is there any other way to do it? I am using Ubuntu GNOME 17.04.  NOTE: I don't want to change the location with dconf-editor as it changes the location of screenshot applet and not of the shortcut I require. How do I prompt for Yes/No/Cancel input in a Linux shell script? What features of Mavericks are beneficial to you? This question serves to share and collect the enhancements that make a large difference to how you use your Mac.   Please post one feature per answer. Please also check to see if your answer has already been posted - duplicate answers will be deleted. To search answers for this question use  (directly from the question page) in addition to your search terms in the search box in the upper right hand corner of this page.  The best answers will not only list a feature, but provide details on how to configure that feature, and provide an image of how to use the feature to be more efficient or effective with Mavericks. Evaluation order of Button's action Why do "df" and "du" commands show different disk usage? I cannot access ubuntu system on my pc, the error message: "the system is running in low graphics mode" and I tried some commands I searched from internet.   I found out a problems seems there is no available disk space. I used "df" and "du" commands to check, the results are as follow:  du -j --max-depth=1 23G  ./home 3.3G ./usr  ......  28G   and  df -Th  filesystem   Type   size    used   available   use%    mounted on  /dev/sda5    ext4   68G     68G    0           100%    /   how would I clean up the system for more disk space? why Booleans slow my computer more than expected What was the origin of Star Trek's Borg? Where can I find documentation on how to use the \nolinebreak command?  The  at  never makes mention of \nolinebreak  The  states that \nolinebreak is implemented by multiplying its input parameter penalty by (-1) and passing the result into \nopagebreak.       I haven't found documentation on how to use the command. Using MAX in SQL Expression of ArcMap ModelBuilder Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future. How can I determine whether a 2D Point is within a Polygon? I'm trying to create a fast 2D point inside polygon algorithm, for use in hit-testing (e.g. Polygon.contains(p:Point)). Suggestions for effective techniques would be appreciated. Cannot find Background Images feature in Blender 2.8 Cannot find Background Images feature to add in a photo in Blender 2.8  Where is it? How to address a professor in letter? How to insert text into a root-owned file using sudo? How can I add a right-click desktop menu entry to create a text file to Gnome 3 in Ubuntu 16.04.02? Proving that cardinality of the reals = cardinality of $[0,1]$ Is it always better to use 'DbContext' instead of 'ObjectContext'? If I seem tired, can I say: "I haven't slept last night"? If not, why have I been told that we use present perfect for actions that have present effects? I made a bootable Windows 7 USB using UNetbootin, and when I try to boot from the bootable USB and select the Default boot menu entry, it gets stuck at the UNetbootin boot menu and it keeps repeating an Automatic boot in 10 seconds... message in an endless countdown loop. Selecting Press [Tab] to edit options and editing the options has no effect. When I go back to the boot menu, it keeps repeating the same countdown loop.   I've tried redoing the UNetbootin extraction, but I still have the same issue.