I'm studying Lie groups and I am in trouble with finding lie algebras of the classical groups. How can I calculate $\mathfrak{sp}(n,\mathbb{C})$ or $\mathfrak{so}(n,\mathbb{R})$ using exp map and tangent space of these groups?
Given a Lie group, how are you meant to find its Lie algebra?  The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them?  My group is the set of all affine maps $x \rightarrow A.x+v$ from $R^n$ to $R^n$ under function composition.
What does this $\asymp$ symbol mean? (subject: analytic number theory) I'm reading a  by Andrew Granville on analytic number theory.  On page 22 of the paper, there appears a strange looking symbol, undefined.  I've circled it in red in the screenshot below.    Since it's not defined in the paper, I'm assuming it must be standard notation.   From the context, I'm assuming it means something like "as compared to", or "with reference to", but that's just a guess.  Can anyone identify the symbol, even better explain what it means and/or provide a reference?  Is there a name to speak the symbol?  Thanks in advance. I have a 3TB Seagate Backup Plus Desktop USB 3.0 drive, which works fine when in its enclosure, but when I get it off its enclosure and directly plug it as an internal SATA drive, it's just not properly recognized (it works again when used in the enclosure). My systems’s motherboard is an , which has two SATA 6.0 Gb/s and four SATA 3.0 Gb/s ports.  When used as an internal SATA drive Windows 7 asks if I want to format the drive, as if it didn’t have a proper filesystem, and if I use the DISKMGMT.MSC tool (typing that in the start menu) I get completely wrong information about the drive.  It says the drive has 3 partitions (349,31GB of RAW data and two unassigned partitions of 1698,68GB and 746,52GB). This information is plain wrong, since the drive, when used in the USB enclosure, works as a single NTFS partition (2794,52GB). How come it’s shown as 3 partitions without filesystem when connected through SATA?!  Is Seagate using some proprietary way of storing the data when using the drive in its provided USB 3.0 enclosure? I didn’t use any kind of encryption. I just plugged the drive with USB 3.0 and started using it straight away, since it apparently came pre-formatted with an NTFS filesystem. How can I align a multiple-line footnote text right to the footnote mark? How can I have the following "footnote layout"?  Some Text Some Text Some Text Some Text² Some Text   2 FOOTNOTETEXT FOOTNOTETEXT FOOTNOTETEXT FOOTNOTETEXT FOOTNOTETEXT   FOOTNOTETEXT FOOTNOTETEXT FOOTNOTETEXT FOOTNOTETEXT FOOTNOTETEXT   My problem is that the footnote text starting with the second line is too far left.  \documentclass{article} \usepackage{blindtext}  \begin{document}     \blindtext\footnote{\blindtext} \end{document} Remove the shortcut of an application that was installed using Wine from the Unity Dash How do I remove an application shortcut from the Unity Dash (not from the Unity Launcher)?  My specific issue is I tried to install a game using Wine a while ago. I then uninstalled the game, then uninstalled Wine. Today I found that shortcuts to the game (and other related content that came with it) remain when I push Super and type the game name. (Wine is not visible though.) Is it possible to have connecting loop lines (like algorithm2e) in algorithmic? I have a quick question about the refraction of light, and I'm sorry if it seems a bit simplistic or even stupid, but I'm still learning.  We know that when light passes through a denser medium, it slows down. However, is there a limit to that slowing down? If no, wouldn't it be possible, with some type of medium we make ourselves (like a crystal), to decrease the light's speed down to a point where we would be able to "see the photons moving forwards"?  I know it is way more complicated than that and that you can't see light unless some other light bounces off it, what I mean is slowing down light to an extreme and unusual speed). That sounds completely absurd but I can't seem to find what is wrong. Can anyone can explain it to me? :) What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$? How can I undo git reset --hard HEAD~1? Is it possible to undo the changes caused by the following command? If so, how?  git reset --hard HEAD~1 I am trying to use a  as input and then creating a query with it - so I need a .    I was surprised to find that it couldn't do the conversion implicitly or explicitly - but I don't even know how I would do this, as the Java API is still fairly new to me. Gauss's mean value theorem problem Suppose that  $f(z)$ is analytic and $f(z)\neq0$  on the disc $|z|&lt;1$ .  Show that for $0&lt;r&lt;1$,  $$ \exp\left[\frac{1}{2\pi}\int_{0}^{2\pi}{\log}|f(re^{i\theta})|\,d\theta\right] = |f(0)| $$  Any other theorems needed to solve this problem? I don't have any idea just with "Gauss's MVT"  Thank you in advanced. One-hot vs dummy encoding in Scikit-learn Determining the dimension of span$\{AB-BA : A,B \in M_{n \times n}(\mathbb R) \}$ Let $S$ be the subspace , of  $M_{n \times n}(\mathbb R)$ (the vector space of all $n \times n$ real matrices ) , generated by matrices of the form $AB-BA$ , where $A,B \in M_{n \times n}(\mathbb R)$ , then how do we prove that $\dim S=n^2-1$ ? The only thing that I can determine is that the trace of all matrices of $S$ is $0$ . Please help Add alias URL for posts timeline 'Sorry' as a replacement for 'Excuse me'? Is it ok to say 'sorry' where you otherwise would use 'excuse me'?  Example: "Sorry, but could you show me the way to the next train station?" Making adult decisions are/is really not fun.     What is the proper answer and why? My gut says "is" but I can't explain why. If granted omniscience and we know that $\beta_1$ in a multiple linear regression model is truly 0, what does that mean in words (and math notation)?  The model is: $Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \epsilon$  Does $\beta_1 = 0$ mean:   Empirically: "If we make infinite, perfect measurements of our data, $\hat\beta_1$ approaches 0 at the limit?" [Math: ??] Formally: "In the true, finite population of our data, $X_1$ has 0 association with $Y$ and their vectors of values are perfectly linearly independent, uncorrelated, and orthogonal?" [Math: ??]   What does $\beta_1 = 0$ mean, precisely? How do I show for an ideal transformer $M^2=L_1L_2$? I've been stuck on this problem for about an hour.     In an ideal transformer, the same flux passes through all turns of the primary and of the secondary . Show that in this case $M^2=L_1L_2$, where $M$ is the mutual inductance of the coils, and $L_1$, $L_2$ are their individual self-inductances.   My first instinct was the following: If we isolate the response of each coil to the other one, we get the equations $N_1\Phi_1=MI_2$ and $N_2\Phi_2=MI_1$, where $\Phi$ is magnetic flux going through a single turn of the coil and $I$ is current. Likewise, if we isolate the response of each coil to itself, we get the equations $N_1\Phi_1=L_1I_1$ and $N_2\Phi_2=L_2I_2$. I would now naturally want to say "Now just multiply $N_1\Phi_1$ and $N_2\Phi_2$ both ways, and get the desired result!" i.e.,  $$(N_1\Phi_1)(N_2\Phi_2)=(MI_2)(MI_1)=(L_1I_1)(L_2I_2)\implies M^2=L_1L_2$$  But that's incorrect, because the response of the coils to themselves and to the other coil are different quantities (i.e. not necessarily equal).  Realizing this, I write down the more general equations.  $$N_1\Phi_1=MI_2+L_1I_1$$ $$N_2\Phi_2=MI_1+L_2I_2$$  Now even with the hypothesis that $\Phi_1=\Phi_2=\Phi$, I still can't see how the identity $M^2=L_1L_2$ follows. Could you help clear up my confusion?  *Taken from D. Griffiths Introduction to Electrodynamics (4th ed.), Problem 7.58. How to convert Local time to UTC time in Delphi? and how to convert it back from UTC to local time? Why does parameter expansion with spaces without quotes work inside double brackets "[[" but not inside single brackets "["? Subject–verb agreement not clear in "Is two bars OK?" Is the following correct?     Can I get some chocolate for myself? Yes, you can but not much.   Is two bars OK?   Is there any rule to explain the sentence above?  The dialogue's taken from Grammar Textbook Round-Up Level #3.