Let $G$ be a group of order $n$. $G$ has one and only one subgroup of order of every divisor of $n$. Then G is cyclic.
A finite group which has a unique subgroup of order $d$ for each $d\mid n$.
Ubuntu 13.10 Macbook Air Mavericks Parallels 9 Black Screen I can install Ubuntu 13.10 in parallels ok and then log in.  Problem is Ubuntu does an update and I get black screen / desktop such that I can't see any app windows that I open.  I've tried a few of the other suggestions and they don't work for me. Chance of selecting the last k pages in correct order from a set of n pages I have a set of documents numbered 1 to n. If I arrange the documents randomly, what is the chance of selecting the last k pages in the correct order if I select them at random from the documents?   Because order is important, I use the following formula:  $\displaystyle\frac{n!}{(n-k)!}$  From this I calculate how many ways there are to select k pages from n pages in the correct order. I need to have the last k pages. Which I can do in only one way. So is the chance   1/$\displaystyle\frac{n!}{(n-k)!}$?   It seems unlikely. It seems to me spontaneously that the chance should be smaller. Extend x axis on graph? I'd like to extend my x axis a little longer. Is it possible to do that?  \centering     \begin{tikzpicture}[scale=1.5]      \begin{axis}[         xlabel=Reynolds Number,ylabel=Coefficient Discharge]     \addplot [color=blue,mark=*] coordinates {         (6485.41491,0.408143397)         (7927.942858,0.407370795)         (8663.115504,0.38550864)         (9219.462371,0.366953133)         (9537.374867,0.346531912)         (10491.11235,0.330115983)         (11087.19828,0.312041115)         (12219.76155,0.313951177)     };         \addlegendentry{1/4}     \addplot [color=red,mark=triangle] coordinates {         (6556.945221,   0.412644978)         (7868.334265,   0.404307857)         (8663.115504,   0.38550864)         (9187.671122,   0.365687778)         (9120.114717,   0.331371141)         (10443.42548,   0.328615455)         (11683.28421,   0.328817519)         (12875.45607,   0.330797337)     };         \addlegendentry{1/2 Lift}     \addplot [color=black,mark=*] coordinates {         (6843.066467,   0.430651305)         (8297.516134,   0.426361012)         (9203.566747,   0.40955872)         (8615.42863,    0.342911376)         (9298.940495,   0.337868614)         (10824.92047,   0.340619673)         (11683.28421,   0.328817519)         (12899.29951,   0.331409925)     };         \addlegendentry{3/4 Lift}     \addplot [color=yellow,mark=*] coordinates {         (6854.988186,   0.431401568)         (8398.850742,   0.431568008)         (9267.149246,   0.412388141)         (8567.741755,   0.341013343)         (9346.62737 ,   0.339601273)         (10753.39016,   0.338368882)         (11897.87515,   0.334857025)         (13030.43841,   0.334779157)     };         \addlegendentry{Full Lift}     \end{axis}% \end{tikzpicture}% In how many ways can you choose $k$ numbers out of $\{1,2,3,\dots,n\}$ so none of them is consecutive? Lie algebra of $\left(\begin{smallmatrix}a & b\\ & a^2\end{smallmatrix}\right )$ in $GL_2(\mathbb{R})$ wrong positioning of figure with tabular in xelatex I recently updated MikTex and it seems Xelatex is not able to place figures correctly anymore. Before updating all figures were shown correctly. I work with Texworks and Miktex 2.9  Here it is the code that I run:  \documentclass[10pt,a4paper]{report} \usepackage{graphicx} \graphicspath{{method1/}{method2/}{dpfig/}}  \begin{document}  \begin{figure}[!htbp] \centering \begin{tabular}{cc} \includegraphics[trim=6cm 2.5cm 6cm 2.5cm,clip=true,width=0.33\textwidth]{figure1.eps}&amp; \includegraphics[trim=6cm 2.5cm 6cm 2.5cm,clip=true,width=0.33\textwidth]{figure2.eps}\\ \end{tabular} \caption{Two figures} \end{figure} \end{document} \end{figure}   With this code, figures are correctly trimmed and the relative position between figure 1 and figure 2 is maintained. However, the position in the document is wrong: figure are placed upper and more right than expected (as the bottom-left corner of the table was placed in the middle of the page). The unusual thing is that this does not happen in draft mode and picture are shown centered as it would be.  I usually compile with Xelatex, Makeindex and Bibtex.  I hope to have been clear.. bye!  EDIT: Sorry, I forgot to add the images I prepared from the file that run that script: I am already a little bit familiar with Linux distros like Debian or Ubuntu (yeah, very similar) but I wanted to try Red Hat based - CentOS 6.2. I have installed it on my Windows 7 host in VirtualBox and tried to play with it a little. I have come across a small problem, namely: the default eth0 interface is down by default. I use the option with NAT (the virtual machine is 'behind' the host). Even if I bring the interface up with ifconfig eth0 up, it does not work right away. I get this after bringing the interface up: eth0      Link encap:Ethernet  HWaddr 08:00:27:0F:00:8A           inet6 addr: fe80::a00:27ff:fe0f:8a/64 Scope:Link           UP BROADCAST RUNNNING MULTICAST MTU:1500 Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:6 errors:0 dropped:0 overruns:0 carriers:0           collisions:0 txqueuelen:1000           RX bytes:0 (0.0 b)  TX bytes:468 (468.0 b)           Interrupt:19 Base address:0xd020  lo        Link encap:Local Loopback           inet addr:127.0.0.1  Mask:255.0.0.0           inet6 addr: ::1/128 Scope:Host           UP LOOPBACK RUNNING  MTU:16436  Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:0 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0           RX bytes:0 (0.0 b)  TX bytes:0 (0.0 b)  virbr0    Link encap:Ethernet  HWaddr 52:54:00:75:C2:9B           inet addr:192.168.122.1  Bcast:192.168.122.255  Mask:255.255.255.0           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:0 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0 [root@centos ~]# _  What should be done more to configure the network on CentOS machine? How is Gauss' Law (integral form) arrived at from Coulomb's Law, and how is the differential form arrived at from that? How to find a computer name in a LAN from the IP address? My network includes machines running Linux and others running Windows. And my machine is running Linux. I have a GTX 660M and Ubuntu 18.04 64-bit.  According to , the latest graphics driver for this card is version 410.66.  The  says  Current long-lived branch release: `nvidia-410` (410.66)   I have nouveau drivers installed at the moment according to lsmod and, apt-get purge 'nvidia*' does nothing.  I have done add-apt-repository ppa:graphics-drivers and apt-get update and a reboot.  Running ubuntu-drivers devices gives this output  == /sys/devices/pci0000:00/0000:00:01.0/0000:01:00.0 == modalias : pci:v000010DEd00000FD4sv00001028sd00000551bc03sc00i00 vendor   : NVIDIA Corporation model    : GK107M [GeForce GTX 660M] driver   : nvidia-304 - third-party free driver   : nvidia-driver-390 - third-party free driver   : nvidia-340 - distro non-free driver   : nvidia-driver-396 - third-party free driver   : nvidia-driver-410 - third-party free recommended driver   : xserver-xorg-video-nouveau - distro free builtin   Note this line:  driver   : nvidia-driver-410 - third-party free recommended   It looks like everything is ready to go to install driver version 410, but autocompleting apt-get install nvidia- indicates the highest available version is nvidia-390, and trying apt-get install nvidia-410 directly fails.  Why does apt not list the version 410 that the PPA and ubuntu-drivers both say is there?  And if I can't apt-get install nvidia-410, how can I install this version? On iPhone it's very easy to clone an app (running multiple copies of the app on the same device); e.g.:    Is it possible to do this on Android? The disk that contains Ubuntu on my computer is 115GB in size.  When the disk was 114.7GBfull.  I started deleting some files here and there to free up around 5GB.  But I noticed the disk was 114GB filled again. I thought it was some cache or swap thingy so didn't give it much thought and went ahead and freed around 40GBby shifting some media files out of the disk.  I have a notification in an hour or so that the disk is full yet again!! 40GB!!! All gone!  So I debugged the problem down to the /var/log/syslog file which was initially definitely less than 20GB. I know this because /var folder was 17GB before I cleared the memory.  /var/log/syslog file is currently 55.9GB!  Could someone be so gracious to explain this anomaly? Is this a bug? Or am I affected with some malware or virus? Comment font sizes are inconsistent on Android Homotopy equivalence of universal cover As part of am exam question (Q21F ), I'm trying to prove that if $X$ and $Y$ are path-connected, locally path-connected spaces with universal covers $\widetilde{X}$ and $\widetilde{Y}$, respectively, then if $X \simeq Y$ then $\widetilde{X} \simeq \widetilde{Y}$.  My attempt might be correct, but it seems too complicated and amongst the jumble I may have made some incorrect assumptions. So what I'm looking for is (a) verification or correction; and (b) ideas as to how I could simplify my argument.    My argument:  Let $p:\widetilde{X} \to X$ and $q:\widetilde{Y} \to Y$ be the covering maps, and let $X \overset{f}{\underset{g}{\leftrightarrows}} Y$ be a homotopy equivalence.  Fix a point $\widetilde{x} \in \widetilde{X}$, let $x=p(x) \in X$ and $y=f(x) \in Y$, and fix $\widetilde{y} \in q^{-1}(\{y\}) \subseteq \widetilde{Y}$. Also let $x' = g(y) \in X$ and fix $\widetilde{x'} \in p^{-1}(\{x'\}) \subseteq \widetilde{X}$.  Define a map $\widetilde{f} : \widetilde{X} \to \widetilde{Y}$ as follows. For $\widetilde{z} \in \widetilde{X}$ let $\widetilde{u}:[0,1] \to \widetilde{X}$ be a path from $\widetilde{x}$ to $\widetilde{z}$. Let $z=p(z)$ so that $u=p\widetilde{u}$ is a path in $X$ from $x$ to $z$. Let $v=fu$, so that $v$ is a path in $Y$ from $y$ to $f(z)$. Lift $v$ to a path $\widetilde{v}$ in $\widetilde{Y}$ with $\widetilde{v}(0) = \widetilde{y}$. Define $\widetilde{f}(\widetilde{z}) = \widetilde{v}(1)$.  Notice that $q\widetilde{v}=v$ so that $q\widetilde{f} = fp$.  Define $\widetilde{g}:Y \to X$ analogously: For $\widetilde{z} \in \widetilde{Y}$ let $\widetilde{a}:[0,1] \to \widetilde{Y}$ be a path from $\widetilde{y}$ to $\widetilde{z}$. Let $z=q(z)$ so that $a=q\widetilde{a}$ is a path in $Y$ from $y$ to $z$. Let $b=fa$, so that $b$ is a path in $X$ from $x'$ to $g(z)$. Lift $b$ to a path $\widetilde{b}$ in $\widetilde{X}$ with $\widetilde{b}(0) = \widetilde{x'}$. Define $\widetilde{g}(\widetilde{z}) = \widetilde{b}(1)$.  Likewise, notice that $p\widetilde{g}=gq$.  Claim: $\widetilde{X} \overset{\widetilde{f}}{\underset{\widetilde{g}}{\leftrightarrows}} \widetilde{Y}$ is a homotopy equivalence.  We have $p\widetilde{g}\widetilde{f} = gq\widetilde{f}=gfp \simeq p$ and $q\widetilde{f}\widetilde{g} = fp\widetilde{g} = fgq \simeq q$.  This shows that $\widetilde{g}\widetilde{f}$ and $\widetilde{f}\widetilde{g}$ are covering translations (deck transformations) and are therefore homotopic to the respective identity maps. So we have a homotopy equivalence.    Any comments would be appreciated. Lost space after removing bootcamp So I removed bootcamp and now there is a partition called "free space" that I can't remove.  Here is my output from diskutil list:    Here is the screenshot from disk utility showing the partitions:    And here is the error I get when I try to remove "free space":    Heres the output when I run   sudo diskutil apfs resizeContainer disk0s2 0 How to find distance from the latitude and longitude of two locations? Normal ordering of the commutator between annihilation and creation operator According to the commutation relation of annihilation and creation operators,   $$[a,a^{\dagger}]=1. \tag{1}$$   I would like to calculate the vacuum expectation value of the normal order of this commutator. We claim that since this commutator is just a c-number, then there is no effect from the normal ordering. We therefore get   $$\langle 0|(:[a,a ^\dagger]:)|0 \rangle=1. \tag{2}$$  However, if we expand the commutator first and do normal order later, we will get something like   $$\langle0|(:[a,a^\dagger]:)|0\rangle=\langle0|(:aa^\dagger-a^\dagger a:)|0\rangle=\langle0|(a^\dagger a-a^\dagger a)|0\rangle=0.\tag{3}$$ Which is contradicting with itself. Does the force of kinetic friction increase with the relative speed of the objects involved? If not, why not? Does the force of kinetic friction increase with the relative speed of the objects involved?    I have heard and read that the answer is no.  This is counter intuitive, and is a big part of why the "Airplane on a Treadmill" question is so interesting.  What phenomena are at work with kinetic friction, and why does it not increase with relative speed? Debugging errors of SAGA interpolation algorithms in QGIS Processing? By integration by parts and the substitution $x = \sin t$ we can easily calculate the integral $\int_{0}^{1} \ln (x+ \sqrt{1-x^2})dx$ which equals to $\sqrt{2} \ln (\sqrt{2} +1) -1.$ I’ve tried to use the same substitution $x = \sin t$ to calculate the integral $ \int_{0}^{1} \frac {\ln (x+ \sqrt{1-x^2})}{x}dx,$ which becomes $$ \int_{0}^{\frac {\pi}{2}} \frac {\ln \sin (t+ \frac {\pi}{4})}{\sin t}dt$$ It seems difficult to solve the particular integral. Any help?