How do I prove the claim of the title?  My definition of Lie Group homomorphism is a smooth group homomorphism.  I have to show that $Df(e)(v)=0$ will imply $v=0$. Viewing this as a derivation, this means $Df(e)(v)(g)=0$ for all smooth functions $g$ on $H$.  Here $f$ is the Lie group homomorphism from $G$ to $H$.
Question statement is as in the title:     $\varphi:G\to H$ is a Lie Group Homomorphism, then $\ker \varphi$ has Lie algebra $\ker D_e \varphi$   Since $\varphi$ is a Lie group homomorphism, in particular it is continuous. As $H$ must be Hausdorff, every point set is closed, and hence $\{e\}$ is closed in $H$, which implies that $\ker\varphi = \varphi^{-1}(e)$ is closed as well.  So $N=\ker\varphi$ is an embedded Lie subgroup of $G$. Therefore, the inclusion $j:N\to G$ is a Lie Group homomorphism, and induces a Lie algebra homomorphism $D_e j :T_e N \to T_e G$. So I only need to show that $D_e j$ is in fact an isomorphism between $T_e N$ and $\ker D_e \varphi$.  But I'm stuck at this point. I know that for $X\in T_e N$, we have, for $f\in C^{\infty}(H)$,  $$ D_e\varphi (D_e j(X))(f)=D_e j (X)(f\circ \varphi)=X(f\circ \varphi\circ j)=0$$  So $D_ej(T_e N) \subset \ker D_e \varphi$.  At this point, I can either show inverse inclusion: that is, for every $X\in \ker D_e \varphi$, we can find $Y\in T_e N$ such that $D_e j (Y)=X$, or for all $f\in C^{\infty}(G)$, $Y(f\circ j)=X(f)$  Or I can somehow show that dimensions of the two spaces are the same.   But I can't proceed from here.  It'd be great if someone could help me
The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$ Conditional Expectaction 3 variables Suppose $X,Y$ and $Z$ are multivariate normal with means and full covariance matrix. The conditional expectation $E(X | Y)$ is well know. What is the conditional expectation of $E(X | Y,Z)$ if $Y$ and $Z$ (and $X$) are correlated? Standard textbooks only seem to cover the case when $Y$ and $Z$ are uncorrelated. The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory:  Lemma If X is a nonempty  with the property that every  of X has an upper bound in X, then X has a maximal element.   Given a orthonormal set $E$ in a Hilbert space $H$, it is apparently possible to show that $H$ has an orthonormal basis containing $E$.   I tried to reason as follows: Suppose $E$ is a finite set of $n$ elements. Then one can number the elements of $E$ to create a totally ordered set of orthonormal elements. Then the span $&lt;E&gt;$ can be identified with $R^n$, where each element $v = v_1 e_1 + v_2 e_2 + ... + v_n e_n$ is identified with the vector $(v_1, v_2, ..., v_n)$. On $R^n$ we have a total order, namely the "lexigraphical order" $(x_1,x_2,...,x_n) \leq (y_1,y_2,...,y_n)$ if $x_1 &lt; y_1$ or if $x_1 = y_1$ and $x_2 &lt; y_2$ or if $x_1 = y_1$, $x_2 = y_2$ and $x_3 &lt; y_3$ and so on. Hence $E$ is a totally ordered subset of $H$ and $H$ is a partially ordered set. However, this set doesn't seem to have an upper bound. The set $E$ does have an upper bound. If we define a total order on $E$ only, then X is a partially ordered set satisfying the criteria so X has a maximal element?   This is as far as I got, and I am not sure the entire argument is correct. I don't see how what kind of maximal element I am seeking, since the orthonormal basis of a Hilbert space can have countably infinity number of elements. How to replace a string with a string containing slash with sed? I am looking for a way to replace a string in a file with a string that contains a slash by using sed.  connect="192.168.100.61/foo" srcText="foo.bar=XPLACEHOLDERX" echo $srcText | sed "s/XPLACEHOLDERX/$connect"   The result is:  sed: -e Expression #1, Character 32: Unknown option for `s' In Group theory proofs what is meant by "well defined" What is exactly meant or required for a mapping to be well defined. I was reading the first Homomorphism theorem () and the first thing the proof does is define a map and find it if its well defined.  Intuitively it makes sense, but what are the requirements for a map to be well defined? For example in the link given, I understand they show one-one relationship as being well defined and later on they again prove its injective.   What have I understood wrongly?  Soham Badass and BAMF are both modern words with approximately the same meaning: "Someone who is awesome to an extreme level, thereby leveraging unquestionable authority." Is there another noun or title that would convey the same level of awe, admiration and toughness that calling someone a badass does, without the vulgarity?  Thanks for any help. :) What is the difference between #include  and #include "filename"? In the C and C++ programming languages, what is the difference between using angle brackets and using quotes in an include statement, as follows?   #include &lt;filename&gt;  #include "filename" Assume you have already completed your undergrad and have been working for a number of years. Does one need to be part of a university or a learning institution to publish papers? Problem with babel and tikz using \draw I'm trying the next code:  \documentclass[pdftex,spanish]{article} \usepackage{babel} \usepackage{tikz}  \begin{document}  Simple arrow:   \begin{tikzpicture}     \draw[-&gt;] (-1.5,0) -- (1.5,0); \end{tikzpicture}  \end{document}   but it doesn't work. It throws the errors:    ! Argument of \language@active@arg&gt; has an extra }. ! Paragraph ended before \language@active@arg&gt; was complete.&lt;to be read again&gt;\par \draw[-&gt;]   What can I do if it works right without the babel package? Must $k$-subalgebra of $k[x]$ be finitely generated? When using sudo to allow edits to files, I regularly get 'permission denied'.  For example, my mouse is jittery and sluggish, so I want to disable polling:  sudo echo "options drm_kms_helper poll=N"&gt;/etc/modprobe.d/local.conf   I'm prompted for a password, and then get:  bash: /etc/modprobe.d/local.conf: Permission denied   So I tried to do a temporary change to disable polling by using:  sudo echo N&gt; /sys/module/drm_kms_helper/parameters/poll   Yet again the system responded with:  bash: /sys/module/drm_kms_helper/parameters/poll: Permission denied   Any ideas? Should I use "who" or "that" when referring to an organisation? I generally use the rule of thumb of using who when referring to a person and that when referring to an object. Example:  Jim is someone who makes me smile. I was nearly knocked out by the ball that hit me.  But what about when we're talking about an organisation?  I am forever indebted to the charities who helped me. I am forever indebted to the charities that helped me.  The former seems somewhat warmer, if that makes sense, although the latter feels more correct. What should I use? And how about this:  I'm really happy with the insurance company who helped me with my claim. I can't stand them, they're the insurance company that ripped me off.  To me the former evokes a team of helpful people while the latter evokes a cold heartless corporate entity. Is this just a matter of style and so should go with my instincts, or should I aim for what is 'correct?' I need to use an alias in the WHERE clause, but It keeps telling me that its an unknown column. Is there any way to get around this issue? I need to select records that have a rating higher than x. Rating is calculated as the following alias:  sum(reviews.rev_rating)/count(reviews.rev_id) as avg_rating Good text on Clinical Trials? I'm an undergraduate statistics student looking for a good treatment of clinical trials analysis. The text should cover the fundamentals of experimental design, blocking, power analysis, latin squares design, and cluster randomization designs, among other topics.   I have an undergraduate knowledge of mathematical statistics and real analysis, but if there's a fantastic text that requires a bit higher level of statistics or analysis, I can work up to it. Let $a,b,c$ be positive real numbers satisfying $$(1+a+b+c)\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=16,$$ then $a+b+c=3$.  I thought about the application of the AM-GM-HM inequality, but in vain. I also thought about splitting 16 into factors and comparing but went nowhere. Any ideas. Thanks beforehand. I am having an issue where DHCP (I though as I read in other similar topics) is clearing the /etc/resolv.conf file on each boot. I am not sure about how to deal with this since the post I have found (,  and some others) are for Debian based distros or other but not Fedora.  This is the output of ifcfg-enp0s31f6 so for sure is DHCP:  cat /etc/sysconfig/network-scripts/ifcfg-enp0s31f6  HWADDR=C8:5B:76:1A:8E:55 TYPE=Ethernet DEFROUTE=yes IPV4_FAILURE_FATAL=no IPV6INIT=no IPV6_AUTOCONF=no IPV6_DEFROUTE=no IPV6_FAILURE_FATAL=no IPV6_ADDR_GEN_MODE=stable-privacy NAME=enp0s31f6 UUID=0af812a3-ac8e-32a0-887d-10884872d6c7 ONBOOT=yes IPV6_PEERDNS=no IPV6_PEERROUTES=no BOOTPROTO=dhcp PEERDNS=yes PEERROUTES=yes   In the other side I don't know if Network Manager is doing something else around this.  Update: Content of NetworkManager.conf (I have removed the comments since are useless)  $ cat /etc/NetworkManager/NetworkManager.conf  [main] #plugins=ifcfg-rh,ibft dns=none  [logging] #domains=ALL   Can I get some help with this? It's annonying be setting up the file once and once on every reboot.  UPDATE 2  After a month I'm still having the same issue where file gets deleted by "something".   Here is the steps I did follow in order to make a fresh test:   Reboot the PC After PC gets restarted open a terminal and try to ping Google servers of course without success:  $ ping google.com ping: google.com: Name or service not known  Check the network configuration were all seems to be fine:  $ cat /etc/sysconfig/network-scripts/ifcfg-enp0s31f6  NAME=enp0s31f6 ONBOOT=yes HWADDR=C8:5B:76:1A:8E:55 MACADDR=C8:5B:76:1A:8E:55 UUID=0af812a3-ac8e-32a0-887d-10884872d6c7 BOOTPROTO=static PEERDNS=no DNS1=8.8.8.8 DNS2=8.8.4.4 DNS3=192.168.1.10 NM_CONTROLLED=yes IPADDR=192.168.1.66 NETMASK=255.255.255.0 BROADCAST=192.168.1.255 GATEWAY=192.168.1.1 TYPE=Ethernet DEFROUTE=yes IPV4_FAILURE_FATAL=no IPV6INIT=no  Restart the network service:  $ sudo service network restart [sudo] password for &lt;current_user&gt;:  Restarting network (via systemctl):                        [  OK  ]  Try to ping Google servers again, with no success:  $ ping google.com ping: google.com: Name or service not known  Check for file /etc/resolv.conf:  $ cat /etc/resolv.conf  cat: /etc/resolv.conf: No such file or directory  File doesn't exists anymore - and this is the problem something is deleting it on every reboot Create the file and add the content of DNS:  $ sudo nano /etc/resolv.conf   Ping Google servers this time with success:  $ ping google.com PING google.com (216.58.192.110) 56(84) bytes of data. 64 bytes from mia07s35-in-f110.1e100.net (216.58.192.110): icmp_seq=1 ttl=57 time=3.87 ms    Any ideas in what could be happening here? Is there a quick way to get to the Control Panel in Windows 8? Obviously in all previous versions of Windows you could simply click START and then select "Control Panel" to get to it, but on Windows 8 there is no such option.  Is there an easy way to quickly get to it without having to press WIN, type "Control Panel", and then press Enter? Software to synchronize two directories (local/remote) I need software that would detect changes in directory (A) every 5 seconds or so and upload/delete changed files from/at (B) remote directory.  I found non-commercial software, . I am wondering if there are any commercial alternatives? Proper usage of "since" and "from" with regard to duration of time Response.Redirect to new window I want to do a Response.Redirect("MyPage.aspx") but have it open in a new browser window. I've done this before without using the JavaScript register script method. I just can't remember how?