Let $G$ be a Lie group with Lie algebra $\mathfrak g$. I'm trying to prove the first two terms of the Baker-Campbell-Hausdorff formula for $\exp(tX)\exp(tY)$ satisfy $$ (\exp tX)(\exp tY) = \exp\big(t(X+Y) + \frac 1 2 t^2 [X,Y] + t^3 \hat Z(t)\big)  $$ for some smooth function $\hat Z: (-\epsilon, \epsilon) \to \mathfrak g$ and some $\epsilon &gt; 0$, for every $X, Y \in \mathfrak g$.   What I've tried: Since $\exp : \mathfrak g \to G$ restricts to a diffeomorphism from a neighborhood of $0 \in \mathfrak{g}$ to a neighborhood of $e \in G$, the map $\phi : (-\epsilon, \epsilon) \to \mathfrak g$ defined by  $$ \phi(t) = \exp^{-1}(\exp tX \exp tY) $$ is smooth for some $\epsilon &gt; 0$. So the problem amounts to proving $\phi''(0) = [X,Y]$ and applying Taylor's theorem. But I'm not sure how to do this without knowing the derivative of multiplication $G \times G \to G$ at general points $(g,h) \in G \times G$. I'm wondering if there's a more clever approach having something to do with the adjoint representations $\mathrm{Ad}: G \to \mathrm{GL}(\mathfrak g)$ and $\mathrm{Ad}_* = \mathrm{ad}: \mathfrak g \to \mathfrak{gl}(\mathfrak{g})$ since $\mathrm{ad}(X)Y = [X,Y]$. Any advice would be appreciated.
Problem 20-9 in Lee's Introduction to Smooth Manifolds (2nd Ed.) asks to prove that for a Lie group $G$, and for $X, Y\in \text{Lie}(G)=\mathfrak g$, there is $\epsilon&gt;0$ and a smooth map $Z:(-\epsilon, \epsilon)\to \mathfrak g$ such that   $$(\exp tX)(\exp tY) = \exp\left(t(X+Y) + \frac{1}{2}t^2[X, Y] + t^3 Z(t)\right)$$   What we want to show is that the first and second term in the Taylor expansion of $\exp^{-1}\left((\exp tX)(\exp tY)\right)$ are $t(X+Y)$ and $\frac{1}{2}t^2[X, Y]$ respectively.  Calculating the first term is easy since we just need to differentiate $\exp^{-1}((\exp tX)(\exp tY))$ with respect to $t$ at $t=0$ once, and this gives the desired result.  But for the second term we need to take the second derivative at which I am stuck.
Can an abstract class have a constructor? Can an abstract class have a constructor?  If so, how can it be used and for what purposes? How do I install the latest stable version of Thunderbird? Reason of existense of the extra space in align environment and removal of it between two align environments This is a follow-up question regarding the extra vertical space in amsmath's align environment. Why when in use align uses so much vertical space in contrast with other environments like for example tabular or itemize?  \documentclass[12pt]{article} \usepackage[top=0.3in, bottom=1.2in, left=0.8in, right=0.8in]{geometry}   \usepackage[fleqn]{amsmath} \usepackage{unicode-math}  \setlength{\mathindent}{0cm} \newcommand{\3}{\vspace{0.3cm}}  \title{} \author{} \date{}  \begin{document}  70  \begin{align*} &amp;e^{jz}=\cos z+j\sin z\\ &amp;\cos z=(1/2)(2\cos z)=\\ &amp;=(1/2)(2\cos z+j\sin z-j\sin z)=\\ &amp;=(1/2)(\cos z+j\sin z+\cos z-j\sin z)=\\ &amp;=(1/2)(e^{jz}+e^{-jz}) \end{align*}  80  \begin{itemize} \item Item 1  \item Item 2  \end{itemize}  90  \begin{tabular}{c|c}  Entry 1&amp; Entry 2  \\  Entry 3&amp; Entry 4 \end{tabular}  \end{document}     Also another image of the huge space between two align environments. Even without the blank lines the space is a lot.    \begin{document}  \begin{multicols*}{2}   \begin{align*} &amp;\text{70}\\ &amp;e^{jz}=\cos z+j\sin z\\ &amp;\cos z=(1/2)(2\cos z)=\\ &amp;=(1/2)(2\cos z+j\sin z-j\sin z)=\\ &amp;=(1/2)(\cos z+j\sin z+\cos z-j\sin z)=\\ &amp;=(1/2)(e^{jz}+e^{-jz}) \end{align*}    \begin{align*} &amp;\text{207}\\ &amp;x(t)=A_{c}\cos \theta(t)\\ &amp;\theta(t)=2\pi f_{c}t+\phi(t)\\ &amp;\phi(t)=K_{p}m(t)\\ &amp;x(t)=A_{c}\cos[2\pi f_{c}t+K_{p}m(t)] \end{align*}   Edit I:  As I mention in a comment I have tried the following code and there is no difference between two align environments, only between text and an align environment. Why doesn't work in the case of the two environments and how can I fix it so to work? Also what is the reason that this space exists?  \setlength{\abovedisplayskip}{0pt} \setlength{\belowdisplayskip}{0pt} \setlength{\abovedisplayshortskip}{0pt} \setlength{\belowdisplayshortskip}{0pt} Why is the subject omitted in sentences like "Thought you'd never ask"? "Thought you'd never ask" is "I thought you'd never ask" with "I" omitted. "Hope this helps" is "I hope this helps" with "I" omitted.   In English grammar, normally every sentence should have a subject, right?   My first thought is that these two examples are so often used that they are like set phrases. But these are not really set phrases. You can alter the words after "thought" and "hope".  Another possible explanation is the tendency to drop the subject if it is the first person pronoun. It seems that in many languages, such as Spanish, Italian and Japanese, the first person subject is usually omitted. Maybe English is going the same way? 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Is using StringBuilder a better choice? dpkg-divert: error: rename involves overwriting `/usr/lib/xorg/protocol.txt' with different file `/usr/lib/xorg/protocol-precise.txt', not allowed I just installed Ubuntu (12.04.4) and I upgraded to 13.10  I have no idea how to fix this :/  $ sudo apt-get install -f  Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be REMOVED:   xserver-common-lts-saucy 0 upgraded, 0 newly installed, 1 to remove and 2 not upgraded. 1 not fully installed or removed. After this operation, 1,718 kB disk space will be freed. Do you want to continue [Y/n]? y (Reading database ... 175174 files and directories currently installed.) Removing xserver-common-lts-saucy ... Removing 'diversion of /usr/lib/xorg/protocol.txt to /usr/lib/xorg/protocol-precise.txt by xserver-common-lts-saucy' dpkg-divert: error: rename involves overwriting `/usr/lib/xorg/protocol.txt' with   different file `/usr/lib/xorg/protocol-precise.txt', not allowed dpkg: error processing xserver-common-lts-saucy (--remove):  subprocess installed post-removal script returned error exit status 2 Errors were encountered while processing:  xserver-common-lts-saucy E: Sub-process /usr/bin/dpkg returned an error code (1) Plotting function defined with Lua doesn't do anything Is volunteer work allowed in the US while visiting under the VWP or a B1/B2 visa? I am going to Santa Monica for 4 weeks to polish my English in July and would like to volunteer at the Planetary Society located in Pasadena.  I am a Swiss citizen and have a biometric passport. However, I don't have any special visa, as there is no visa needed for the school I am attending and using the Visa Waiver Program, I am allowed to stay up to 90 days in the US.  Am I allowed to do volunteer work during my stay? Is there a way to show the console in a Windows application?  I want to do something like this:  static class Program {     [STAThread]     static void Main(string[] args) {         bool consoleMode = Boolean.Parse(args[0]);          if (consoleMode) {             Console.WriteLine("consolemode started");             // ...         } else {             Application.EnableVisualStyles();             Application.SetCompatibleTextRenderingDefault(false);             Application.Run(new Form1());         }     } } Relevant:  and   The answers in the linked question don't quite help me. Specifically, what happens with this case:  My three pens are green. Each of my three pens are green. Each of my three pens is green.  Part of the confusion seems to stem from the inclusion of &quot;three&quot;:  My pens are green. Each of my pens are green. Each pen is green.  There doesn't seem to be a way to rewrite the last example with &quot;three&quot;:  Each three pen is green.  The more common application would be:  All three pens are green.  But in the event that I want to say, &quot;Each of my three pens are green&quot; which is correct? And, in the event that the correct sentence is &quot;each of my three pens is green&quot; why does prepending &quot;each of&quot; to the beginning of &quot;my three pens are green&quot; change the verb? Deleted question showing up in 'newest questions' tab What's the subject of "There is my biscuit!" ? And how about "There is one biscuit left"? What's the subject, grammatically speaking, of these sentences?   There is my biscuit! My biscuit is there! There is one biscuit left.   I don't really know how to analyze these. 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