What about the index of this subgroup?
Poincaré's theorem about groups
Non-algebraically closed field in which every polynomial of degree $<n$ has a root My problem is to build, for every prime $p$, a field of characteristic $p$ in which every polynomial of degree $\leq n$ ($n$ a fixed natural number) has a root, but such that the field is not algebraically closed.  If I'm not wrong (please correct me if I am) such a field cannot be finite, by counting arguments. But on the other hand, the union of all finite fields (or of any ascending chain of finite fields) of characteristic $p$, which is what I get if I start with $F_p$ and add a root to each polynomial of degree $\leq n$ in each step, is the algebraic closure of $F_p$, hence algebraically closed. I don't see how I can control this process so that in the end I get a field that is not algebraically closed.  Any hint will be welcome. Thanks in advance. Calculating size of particular area below buffer in QGIS? I have a point shapefile of villages where data was collected in Northern Kenya. I created a buffer around those villages.   The second vector layer contains polygons defining areas according to vegetation type. So for example, a forest might be surrounded by a narrow ring of woodland in an area which is mainly grassland. I'm trying to obtain a percentage of each vegetation type within the buffer per village.   I'm using QGIS 1.7 Almost every time I turn my XBox on and start up Fallout 4, I am unable to use the "Continue" option because the most recent save is corrupt.  I instead use the "Load" option. Here, I can see that the last save I manually performed is perfectly fine, and load it with no problem. However, there is an additional, corrupt save with the same time stamp immediately after it.  This doesn't happen quite every time, but often enough that I'm surprised when there's no corrupt save waiting for me to delete it.  I've tried a few things to solve this, including exiting to the game's main menu before turning the XBox off, going out to the guide before turning the XBox off, turning it off immediately after saving, and waiting a while after saving.  Why do I get all these corrupt saves? Do others have this problem? How do I fix it? How do I insert a tie (~, non-breaking space, tilde) under expl3 syntax? I think the title says all I want: I want to add context menu items for a specific file type like the yellow marked ones in the screenshot. Which registry changes do I have to make? In my case I want to assign ir.exe to .rb files. How can I convert from hex to base64? Can anyone recommend a straightforward way/tool to convert hex to base64?  I'm using Linux and OS X. The spells that we know about in the Harry Potter world all seem to be either in Latin, derived from Latin, or resembling Latin.  This makes sense for the European World, but there were likely witches and wizards in Africa, Asia, and the Americas.  These cultures would have developed independently, and the magical people in them would likely have still found ways to cast their spells.  Do cultures with languages that did not derive from Latin use the same words and phrases that are related to Latin for their spells?  Or did they have other phrases based on the languages in their cultures? Chrome's default search engine is .  How we can force chrome to use  without https?  I need a solution that allows proxies to rewrite my URLs, but https currently makes this impossible... I hope this isn't a silly question but I've been watching TNG and this bothers me.  The transporter 'pad' area is relatively small.  As a guess, maybe 12 people can fit on it.  But the vast majority of times it's being used by 2-3 people.  However, the transporter is able to beam people from anywhere on the ship to anywhere (within some distance I presume).  It's fairly common for them to perform a transport directly to the bridge.  So, why is there an actual room?    If a three man away team is to beam down to the surface of a planet - why do they have to walk all the way to the transporter room, then stand on the pad to get down there?  Why not simply beam them from where they are, to the surface.  The ship's computer is able to locate anyone on board. Why is ANOVA taught / used as if it is a different research methodology compared to linear regression? ANOVA is equivalent to linear regression with the use of suitable dummy variables. The conclusions remain the same irrespective of whether you use ANOVA or linear regression.   In light of their equivalence, is there any reason why ANOVA is used instead of linear regression?   Note: I am particularly interested in hearing about technical reasons for the use of ANOVA instead of linear regression.   Edit  Here is one example using one-way ANOVA. Suppose, you want to know if the average height of male and females is the same. To test for your hypothesis you would collect data from a random   sample of male and females (say 30 each) and perform the ANOVA analysis (i.e., sum of squares for gender and error) to decide whether an effect exists.  You could also use linear regression to test for this as follows:  Define:  $\text{Gender} = 1$ if respondent is a male and $0$ otherwise. $$ \text{Height} = \text{Intercept} + \beta * \text{Gender} + \text{error} $$ where: $\text{error}\sim\mathcal N(0,\sigma^2)$  Then a test of whether $\beta = 0$ is a an equivalent test for your hypothesis. diameter on a compact metric space I have troubles showing the following:  Let $(X,\rho)$ be a compact metric space and $F \subset X$ a closed subset. Prove that if diam $F &lt; \infty$, then there exist $x_{0}, y_{0} \in F$ such that diam $F= \rho(x_{0},y_{0})$.  It looks so trivial yet hard to find the trick. Count and limit the number of files uploaded (HTML file input) I know how to define a package or a class, but how do I make it available (to myself) at all times?  Say I have the package file commonstuff.sty (or myprettyclass.cls), that I want to be able to include in any .tex file I create on my computer. Where do I put it? Do I need to run some installer on it, or is it enough to keep it in a certain folder?  Note: I know I can just place it in the same folder as the .tex file, but that's not what I'm after here. I'm looking for a way to "install" the package centrally on my computer. I have a machine running Ubuntu which I SSH to from my Fedora 14 machine. I want to forward X from the Ubuntu machine back to Fedora so I can run graphical programs remotely. Both machines are on a LAN.  I know that the -X option enables X11 forwarding in SSH, but I feel like I am missing some of the steps.  What are the required steps to forward X from a Ubuntu machine to Fedora over SSH? Advice/literature on combining items with different response scales into composite scales? Let's say I have some self-report items measured on a 5-point Likert scale (Strongly Disagree to Strongly Agree) and other items measured on a 4-point Likert scale (Never, Rarely, Sometimes, Often). Can anyone point me to literature (or practical advice) on combining these items into a composite scale? Let's assume for the sake of argument that we have some empirical evidence that items should be combined.  Some ideas:  1. Sum raw scores  Pro: Easy  Con: Max responses on the 5-point scale (originally 4's) drive up the total scale score more than max responses on the 4-point scale (originally 3's).  2. Rescale and sum  Put all of the items on a 0-1 scale and sum. So the 4-point items (0,1,2,3) would be multiplied by (4/3)/4 and the 5-point items (0,1,2,3,4) would be multiplied by 1/4, resulting in possible values of (0,.33,.66,1) and (0,.25,.50,.75,1), respectively. This way max responses on the 5-point scale (originally 4's) would not drive up the total scale score more than max responses on the 4-point scale (originally 3's).   Pro: Items would have equal weighting. (could be a con, depending on your perspective).  Con: Ignores differences in variability between items on different metrics?   3. Standardize and sum  A related approach would be to standardize all of the items (z score) and then sum.  Pro: Addresses differences in variability between items on different metrics  Con: Total scale score becomes less interpretable and sample-specific. The latter makes it hard to benchmark as a measure to be used in other settings/other samples.  4. PCA or other data reduction   4a. EFA to get factor loadings. Multiply scaled items by factor loadings.  4b. PCA to get score of first principal component.  Pro: Items weighted by influence.  Con: Same as #2. EFA-derived scores could vary a lot depending on rotation/extraction choices. Some would not advise on ordinal data.    Overall: I like #2 because it seems easier to compare results across different samples. Thoughts? Alternative ideas or concerns about the ideas presented? How to derive this version of variance formula A probably very easy computational question, but I don't really understand how it's done:  I try to compute the point estimates of a normal distribution as $N \sim (\beta, \sigma)$. Using the method of moments, we get:  $$\frac{1}{n} \sum_{i=1}^{n} X_{i} = \hat{\beta}$$  $$\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} = \hat{\sigma}^2 + \hat{\beta}^2$$  Solving the two equations for the two parameters, we get:  $$\hat{\beta} = \overline{X} $$   and   $$\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^{n} (X_{i} - \overline{X}_n)^2 $$  My question is: How do we get to the term of $\hat{\sigma}^2$? If I try to compute it, I get:  $$\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^{n} X_{i}^2 - (\frac{1}{n} \sum_{i=1}^{n} X_{i} )^2 $$  Can someone show me how to simplify the last term to what is stated above and what "rule" or law" is used there? Since it is used so often, I really wanna understand this. I am currently using a macbook pro but the operating system is debian 8.2 not os x  I am using a macbook pro but there's this one device that's just draining a lot of power. This USB device: Card Reader (Apple)   32.5 W    100.0%                      Device         USB device: Card Reader (Apple)   2.60 W    100.0%                      Device         Radio device: btusb   209 mW     19.7 pkts/s                Device         Network interface: wlan0 (wl)   This device really drains my battery, I noticed that my laptop wouldn't even get 2 hours when not plugged in, I installed powertop. Made all the recommended changes but this device, I can't find out how to disable it. I don't even use that reader and forgot that it was there.  I configured all my devices to use limited power in my rc.local  echo '0' &gt; '/proc/sys/kernel/nmi_watchdog' echo 'min_power' &gt; '/sys/class/scsi_host/host0/link_power_management_policy' echo '1' &gt; '/sys/module/snd_hda_intel/parameters/power_save' echo '1500' &gt; '/proc/sys/vm/dirty_writeback_centisecs' echo 'auto' &gt; '/sys/bus/usb/devices/2-4/power/control' echo 'auto' &gt; '/sys/bus/usb/devices/1-12/power/control' echo 'auto' &gt; '/sys/bus/usb/devices/1-8.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1f.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:01:00.1/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.4/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.2/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1b.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:16.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:14.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:01.1/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:01.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:01:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:08:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:06.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:05.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:04.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:03.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:06:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:05:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:04:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:03:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1f.0/power/control'   How can I get this device to act sanely or disable it completely because I  haven't used it. I have a material assigned to a few test objects with an image sequence as a texture. If I view the scene in render preview, and scrub the timeline, (or render a sequence,) the texture behaves as expected.  However, with the viewport set to 'texture' or 'material' I only see the first frame of the sequence, wherever I am on the timeline. 'Auto Refresh' is set in the cycles texture node. Is this the expected behaviour? If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.  I'm wondering if you can help me with this one? Allow linking to specific revision of revision history showing diff Currently, the only way to link to a revision (clicking "link"), you just get  which shows what the post looked like at that revision:        Personally, I rarely use this and haven't ever found myself in a scenario (as far as I remember) where I would prefer the way it currently works to what I'm suggesting here.  Could we:   Add another link, or Change the current link   Such that it:   Displays the diff on a separate page (I'd prefer this, since the highlight and scroll thing doesn't always work properly - although perhaps it has something to do with loading images, so it might not apply here, or, if it doesn't, we could fix that instead), or          Highlights and scrolls to a diff, like what is done when you link to an answer ()