Let $G$ be a lie group and $\mathfrak{g}$ it's associated lie algebra.I can quite easily show that if there is a Lie group representation $\rho_{G}$ from $G$ into $L(G)$:  $$ \rho_{G}: G \rightarrow \mathfrak{g} \;  ; g \rightarrow Ad(g)  $$  Such that for any $X \in \mathfrak{g}$,  $Ad(g)X:=gXg^{-1}$  Then we have a corresponding lie algebra representation $\rho_{\mathfrak{g}}$:  $$ \rho_{\mathfrak{g}}: \mathfrak{g} \rightarrow End(\mathfrak{g}) \; ; X \rightarrow ad(X)$$  Such that $ ad(X)Y = [X,Y]$   But why is it true that the generators of this are the structure constants? (i.e the basis of the adjoint algebra is the structure constants?)    Since $ad$ is just a LA homomorphism, if $T_{\alpha}$ is a generator of $\mathfrak{g}$ then, $ad(T_{\alpha})$ must be a generator of $End(\mathfrak{g})$? But $ad(T_{\alpha}) = [T_{\alpha},-]$ Clearly I'm seriously confused about something here...  EDIT   I think my confusion has something to do with a lack of understanding of dual spaces.(?)   The point is, $ad(T_{\alpha}) = [T_{\alpha},-]$ is indeed the corresponding basis operator in the adjoint representation However that operator can be defined (in a certain basis) as a matrix - and the natural basis to define that matrix in is our original $\{T_{\rho}\}$. In which case:     $$ ad(T_{\alpha})_{\text{Basis of}\mathfrak{g}} = \begin{pmatrix} [T_{\alpha},T_{1}].... [T_{\alpha},T_{n}] \end{pmatrix} $$    So that the components of $ad(T_{\alpha})$ in this basis are indeed the structure constants - although technically, it is more correct to say that $ad(T_{\alpha})$ is the abstract generator?
Given the Lie algebra, what is the systematic way to construct the matrix representation of the generators of the desired dimension? I ask this question here because it is the physicists for whom representation of groups is more important than mathematicians.   Let us, for example, take $SU(2)$ for concreteness. Starting from the generic parametrization of a $3\times 3$ unitary matrix $U$ with $\det U=1$, and using the formula of generators $$J^i=-i\Big(\frac{\partial U}{\partial \theta_i}\Big)_{\{\theta_i=0\}}$$ one can find the $3\times 3$ matrix representation of the generators.  However, I'm looking for something else.    Given the Lie algebra $[J^i, J^j]=i\epsilon^{ijk}J^k$, is there a way that one can explicitly construct (not by guess or trial) the $3\times 3$ representations of $\{J^i\}$? Will the same procedure apply to solve other Lie algebras appearing in physics such as that of $SO(3,1)$ (or $SL(2,\mathbb{C})$)?
How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$ If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$? So the question is really hard I think. I tried using a simple way by calculating the probability of each combination that makes a sum divisible by six, but it would take forever. Does anyone have any ideas?  Suppose that we roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by six? I have a brand new 32GB flash drive that was bought from Hong Kong but manufactured in China.  When I plug it in, it's detected, and when I examine the properties of the drive the size is reported to be 31.2 GB.  Windows forces me to format it but I have not manged to actually do that.  I then I browsed the internet and a forum told me to change the letter to L. After I did that I searched many programs that can help me format it, but when I checked it back, it is 4MB now.  Any idea what's happening with my drive? My initial problem was that my computer wouldn't let me login to my account. It seems to shut down after I enter my password. I can login as a guest.  I found fixes but I can't remember / figure out my login for access after I hit Ctrl+Alt+F1 and most of the fixes for finding your login username depend on you having access to your account.  I thought the login was the name which appears on the screen when I start the computer normally, but this has not worked, is there another way to find out my username while being effectively locked out of my root-user profile?  I am also assuming that the password is the same one I always use.  Conversely, if the name which appears on the screen at startup and my normal password should have worked, is there another explanation for why I am not able to login on the black command screen. How many Marvel Earths (Universes) are there? I know that there are at least 3 Marvel Universes:  Marvel Cinematic Universe -    Has all of the main ongoing Marvel movies    (1, 2)  (1, 2, 3)  (1, Age of Ultron) etc    Marvel's Ultimate Universe -    Has the other Marvel movies    (1, 2, 3, Origins: Wolverine, First Class, The Wolverine, Days of Future Past)  (1, 2)  etc       Main comic book continuity This is the big one, it contains         etc     Then there are apparently off-shoot Earths, continuities which deviate from the main continuity (Earth 616). How many Marvel Earths (Universes) are there?   Further Reading: Can I find out which tags I have created? I am asked to solve the following question using central limit theorem. In an election between two candidates, A and B, one million individuals cast theirvote. Among these, 2000 know candidate A from her election campaign and vote unanimously for her.The remaining 998000 voters are undecided and make their decision independently of each other byflipping a fair coin.Approximate the probability pA that candidate A wins up to 3 significant figures.  It's easy to solve directly. PA = $\frac{0.5(10000-2000)+2000}{10000}$ = 0.501. However, I am quite confused about how to solve this problem by central limit theorem. Personal Hotspot not working over Wi-Fi on iOS 13 After upgrade to iOS 13 from 12.4, I noticed that personal hotspot isn't working anymore over Wi-Fi (via USB is ok).  Tested on two computers with Win7. The PC can connect to Wi-Fi, but no valid IP (IPv4) is assigned and no connection is working to internet. Reason for strange "canyons" of dust on the cooling platform I took apart my cooling platform Something like this :  And I found something like this inside :       I wonder why around the hollows are the "canyon like" structures.  PS: I also drop this platform on the floor, maybe this could be a reason ?   EDIT :  Everybody seems (except Brandon Enright answer) to ignore the word "dust" in this question. Those structures are not in the plastic itself. It is dust ON the plastic.  Proof :     Completely wiped out with my finger : How to understand degrees of freedom? How can you print a variable name in python? Say I have a variable named choice it is equal to 2. How would I access the name of the variable? Something equivalent to  In [53]: namestr(choice) Out[53]: 'choice'   for use in making a dictionary. There's a good way to do this and I'm just missing it.  EDIT:  The reason to do this is thus. I am running some data analysis stuff where I call the program with multiple parameters that I would like to tweak, or not tweak, at runtime. I read in the parameters I used in the last run from a .config file formated as  filename no_sig_resonance.dat  mass_peak 700  choice 1,2,3   When prompted for values, the previously used is displayed and an empty string input will use the previously used value.  My question comes about because when it comes to writing the dictionary that these values have been scanned into. If a parameter is needed I run get_param which accesses the file and finds the parameter.   I think I will avoid the problem all together by reading the .config file once and producing a dictionary from that. I avoided that originally for... reasons I no longer remember. Perfect situation to update my code! Which is correct?        She was told that it was against the law to steal.      Or        She was told that it is against the law to steal.        Background information: The original context is that I was relating an incident via email, where my colleague told a customer that some of our actions were part of the organisation's policy. The policy is still valid, but the statement I used to relate the incident in my email went like this:    "Mr Jay was informed that it was our policy to do this."   And I received feedback that it should go like this instead:    "Mr Jay was informed that it is our policy to do this." How do I study the "correlation" between a continuous variable and a categorical variable? Prove that negative absolute temperatures are actually hotter than positive absolute temperatures Could someone provide me with a mathematical proof of why, a system with an absolute  (such that of a spin system) is  than any system with a positive temperature (in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system). I am looking for a reference which has a self-contained (elementary, that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does anyone know such a reference?   (Note: I have looked at the similar questions on this website but don't find them particularly useful). How should one interpret the comparison of means from different sample sizes? Take the case of book ratings on a website. Book A is rated by 10,000 people with an average rating of 4.25 and the variance $\sigma = 0.5$. Similarly Book B is rated by 100 people and has a rating of 4.5 with $\sigma = 0.25$.   Now because of the large sample size of Book A the 'mean stabilized' to 4.25. Now for 100 people, it may be that if more people read Book B the mean rating may fall to 4 or 4.25.   how should one interpret the comparison of means from different samples and what are the best conclusions one can/should draw?    For example - can we really say Book B is better than Book A. Ubuntu 20.04 on DualBoot fail 90% of tries - Initramfs unpacking failed: Decoding failed I just installed Ubuntun 20.04 on a Dell Latitude 7400 on DualBoot. I had to remove the RST as mentioned on this post :  It works but now I need to test 5 or more times to boot. After the grub screen, most of the time i have a black screen. Sometimes I have this error : Initramfs unpacking failed: Decoding failed And then it starts.  here is a file called initramfs.conf cat /etc/initramfs-tools/initramfs.conf MODULES=most  BUSYBOX=auto  COMPCACHE_SIZE=&quot;&quot;  COMPRESS=lz4   DEVICE=  NFSROOT=auto RUNSIZE=10% This is something that has long bothered me, and I have asked a few physicists and chemists and never gotten a very satisfying answer. Why are particles charged? And I'm not asking (and this is the answer you often get) why some molecule might be negatively charged because the answer to that is simply is that there was one more negatively charged particle than positively charged particle. That doesn't answer my question, however, of why there are charged particles at all?  So, on a fundamental level, what makes one particle charged and another one neutral.  For instance, protons and neutrons have almost exactly the same mass, which I'm assuming that means they have very similar internal compositions, yet one is positively charged and the other neutral. What is going on that makes these particles different?  On a similar note, and I'd really like to know this, how is it that a proton and an electron have exactly equal and opposite charge, yet an electron is about 1000 times less massive than a proton?  I have particular trouble with this question because if the answer were something along the lines of "particle X which composes protons all have partial positive charges of n amount..." I would just ask why particle X has any charge and we'd be in the same predicament.  I'm sure there is a satisfactory answer to my question, I just can't find it and don't know what resources are reliable or detailed enough to be satisfactory. how do we know the BBP formula for $\pi$ is valid? I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we can verify for every $\pi$ digit discovered till now with a "standard formula", how do we know that the formula is correct after those digits?  Also, what's the implication of these kind of BBP formulas for other "divine"/fundamental constants? can we learn anything else about the value of such constants and our universe through these? Any insight into the "supreme fascist" mind? I have a button created by my script. I wonder how I can display what the system console displays but inside blender's console. CoDEmanX did this by running a script inside it as shown , but I want to display what happens in the system console inside blender's console.  My .blend file can be found here: