Let $G$ be a group and (the center of $G$) $Z(G)=\{e\}$. Prove that $Z(\mathrm{Aut}(G))=Z(\mathrm{Inn}(G))=\{I_G\}$ Since  $\mathrm{Inn}(G)\leq \mathrm{Aut}(G)$, it seems that we only need to prove $Z(\mathrm{Inn}(G))=\{I_G\}$, but I do not know how to continue. Appreciate your help!
I am trying to show that if for a group $G$ we have $Z(G)=1$, then $Z(\operatorname{Aut}(G))=1$.  I have tried everything I can think of, but to no avail.  The first part of the problem was to show that $\operatorname{Inn}(G) \triangleleft  \operatorname{Aut}(G)$ - does this have any connection?  I couldn't find one.  I know that $\operatorname{Inn}(G) \cong G/Z(G)$, so in this case $\operatorname{Inn}(G) \cong G$ and $Z(\operatorname{Inn}(G))=1$.  Is there any way to get from this to $Z(\operatorname{Aut}(G))=1$?  How do we go about showing this?  Thanks.
Introduction to Statistical Learning defines the F-statistic as follows: $$\frac{(TSS - RSS) / p}{RSS / (n - p - 1)}$$ I am trying to interpret this formula intuitively - the numerator looks like the $ESS$ per regressor, and the denominator looks like the $RSS$ per observation. This does not seem like an apples to apples comparison; can anyone explain why it makes sense?  They also say that the expectation of the denominator is equal to the variance of the irreducible error if the linear model assumptions hold (I get this, as the denominator is really the Residual Standard Error, which is an unbiased estimator). They also say the numerator is equal to the variance of the irreducible error if the null hypothesis is true. Therefore the F-statistic will be close to 1 if the null hypothesis is true.  But if you let p = 1, that is, apply the F-statistic to single linear regression, it becomes:  $$\frac{TSS - RSS}{RSS / (n-2)}$$  According to the book, if the regressor has no explanatory power, the F-statistic should be close to 1. But if you imagine a data set where the coefficient on X is 0 (i.e. no explanatory power), $TSS$ will be equal to $RSS$, so the numerator and hence the F-statistic should be 0, not 1 as they claim. What is going on?  Furthermore, if you accept that the F-statistic is 1, as they claim, then $(TSS - RSS) = RSS / (n-2)$, which means $ESS = RSS / (n-2)$. If you think of the F-statistics as comparing explained vs. unexplained variation, this does not seem like a fair comparison since this decomposition makes it the ratio of TOTAL explained variation summed across ALL observations vs. the unexplained variation PER observation. Again, what am I missing?  I am just trying to make sense of it in a layman way, apologies if I am missing something obvious. How to trim whitespace from a Bash variable? I have a shell script with this code:  var=`hg st -R "$path"` if [ -n "$var" ]; then     echo $var fi   But the conditional code always executes, because hg st always prints at least one newline character.   Is there a simple way to strip whitespace from $var (like trim() in )?   or   Is there a standard way of dealing with this issue?   I could use  or , but I'd like to think there is a more elegant solution to this problem. Compare means of group with a mean of a subgroup Friendly way to make previous large calculation results reusable in a new session If $G$ is a compact topological group, how to show that a finite index subgroup of $G$ is open ? I really don't know where/how to start...  PS : I precise that by "compact" I mean that it is hausdorff and that any recovering of $G$ by open sets has a finite sub-recovering (sorry if I wasn't unclear) I'm currently working on some logging code that supposed to - among other things - print information about the calling function. This should be relatively easy, standard C++ has a type_info class. This contains the name of the typeid'd class/function/etc. but it's mangled. It's not very useful. I.e. typeid(std::vector&lt;int&gt;).name() returns St6vectorIiSaIiEE.  Is there a way to produce something useful from this? Like std::vector&lt;int&gt; for the above example. If it only works for non-template classes, that's fine too.  The solution should work for gcc, but it would be better if I could port it. It's for logging so it's not so important that it can't be turned off, but it should be helpful for debugging. How to set system-wide environment variables on OS X Mavericks Superluminal neutrinos I was quite surprised to read this all over the news today:     Elusive, nearly massive subatomic particles called neutrinos appear to travel just faster than light, a team of physicists in Europe reports. If so, the observation would wreck Einstein's theory of special relativity, which demands that nothing can travel faster than light.   &mdash;  Apparently a CERN/Gran Sasso team measured a faster-than-light speed for neutrinos.   Is this even remotely possible? If so, would it be a real violation of Lorentz invariance or an "" effect?   The ; a webcast is/was planned .  News conference video I have to import large Shapefiles (> 1 million records) into PostGIS, and I have been wondering about the best way to do it.    In my question I used the word "hack", instead of tool, on purpose because I think this is not so much a matter of which tool, but which set of steps, or configuration settings to use. So far, I have tried the  plugin (QGIS), the  Postgis tool and the GDAL  tool. You can view my full review on  post. So far, I find all of them really unresponsive, when dealing with a large dataset. I was wondering if someone experienced a similar issue, and if you could share something about the approach. Is the War Machine armour simply "the Iron Man armour with big guns"? What are the differences between Iron Man and War Machine armours ?  In the movies, it seems that they are the same, except the War Machine armour features more heavy weapons. But if they are the same, why wouldn't Tony Stark use the War Machine armour?  Do they have different characteristics (like strength, resistance, flight time...)?  I am looking for answers from the movies and/or the comics. How to evaluate the integral $$\int_{25\pi / 4}^{53\pi / 4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx\ ?$$ When I look at the definition given from the Mac OS X Dictionary (I have set American English as interface language, and the dictionary used is then the New Oxford American Dictionary), I read:     the /ði/ /ð(ə)/ /ði/ [called the definite article]   adjective   1. denoting one or more people or things already mentioned or assumed to be common knowledge: what's the matter? | call the doctor | the phone rang. Compare with a.   • used to refer to a person, place, or thing that is unique: the Queen | the Mona Lisa | the Nile.   • informal denoting a disease or affliction: I've got the flu.   • (with a unit of time) the present; the current: dish of the day | man of the moment.   • informal used instead of a possessive to refer to someone with whom the speaker or person addressed is associated: I'm meeting the boss | how's the family?   • used with a surname to refer to a family or married couple: the Johnsons were not wealthy.   • used before the surname of the chief of a Scottish or Irish clan: the O'Donoghue.   I have never thought of an article as adjective. Is normally an article defined as adjective? Beamer itemize not aligned when in blocks While making a poster, I've got two columns with blocks with itemize inside them. MWE:  \documentclass[final]{beamer} \mode&lt;presentation&gt;{}  \usepackage{times} \usepackage{amsmath,amssymb} \usepackage[english]{babel} \usepackage[latin1]{inputenc}  \begin{document} \begin{frame}[t]{}   %\leavevmode \begin{columns}[t, onlytextwidth] \begin{column}{.5\linewidth} \begin{block}{Contributions} \begin{itemize} \item First Item \item Second Item \end{itemize} \end{block} \end{column} \begin{column}{.5\linewidth} \begin{block}{Approach} \begin{itemize} \item First Item \item Second Item \end{itemize} \end{block} \end{column} \end{columns}  \end{frame} \end{document}   My local installation and ShareLaTeX would misalign the bullets(baselines added by me):   This problem seems similar to:        But this time it happens when itemize is inside the block. If one removes the blocks it works fine.  Am I missing something or does it look like a Beamer bug?  Thanks! I'm a LaTeX noob, sorry if I missed something stupid. Headers of \chapters* take the headers of their previous \chapter When using book.cls and using \chapter* after a \chapter, the header used in \chapter* is wrong. An example is pasted below.   Is there any known workaround for that? I have tried quite some tricks without success.   \documentclass[]{book}  \begin{document}  \chapter{First Chapter} aslkjdf  \chapter*{Unnumbered Chapter} The GNU General Public License is a free, copyleft license for software and other kinds of works.  The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program--to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.  When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things.  To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others.  For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.  Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.  For the developers' and authors' protection, the GPL clearly explains that there is no warranty for this free software. For both users' and authors' sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions.  Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users' freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.  Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.  The precise terms and conditions for copying, distribution and modification follow.  \end{document} A uniformly continuous function maps bounded set to bounded sets I am trying to prove the following:  If $A\subset\mathbb R$ is bounded and $\,f:A\to \mathbb R\,$ is uniformly continuous, then $f[A]$ is bounded.  Could you check my proof?  Let $A \subseteq [-K,K]\subseteq \mathbb R$. Let $\varepsilon = 1$. If $f$ is uniformly continuous there is $\delta$ wuth $|x-y| &lt;\delta$ imply that $|f(x)-f(y)|&lt;1$ for all $x,y\in A$. Let $a \in A$. Then because $A$ is bounded there is a finite number of balls $B(a_n,\delta)$ that cover $A$. Let the number be $N$. Then $f(a)-N \le f(x) \le f(a) + N$ for all $x\in A$. How to remove uninstalled snaps from cache? Inverse metric in linearised gravity How can this English sentence be translated into a logical expression? ( Translating " unless") You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.   Let:   $P$ stands for "you can ride the roller coaster" $Q$ stands for "you are under 4 feet tall" $R$ stands for "you are older than 16 years old"   Is this logical expression correctly translated?  $$P \rightarrow (Q \wedge R)$$ For some reason my Steam just broke today after working perfectly since I installed it a few days ago.  It was originally installed via the deb from the Steam site. So I uninstalled that using the Ubuntu Software Center. Then I installed steam-launcher. When I run it from the menu it does nothing at all. When I go through terminal it gives this message, no error:  $ steam/usr/bin/steam:  line 196: /home/andrew/.local/share/Steam/steam.sh: Success   So I purge-removed it and removed the .steams directory from my home directory but on re-installation its the same story.  I am using Xubuntu 12.10 64bit.  It appears that steam .local/share/Steam/steam.sh is blank. What is it meant to have in it? I am so confused with how reinstalling it doesn't fix it. How do I escape a single quote ( ' ) in JavaScript?