Evaluating the limit of $\left(\frac{\sin x}{x}\right)^\frac{1}{x^2}$ for $x\to 0^+$
Evaluating $ \lim_{x \rightarrow 0}\left (\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$
Example where union of increasing sigma algebras is not a sigma algebra Constructing an explicit isomorphism between finite extensions of finite fields Suppose $K$ is a finite field, $K = \mathbb F_{p^s}$. If we take an irreducible polynomial $f$ of degree $d$ over $K$, then the splitting field $L$ of $f$ is $K(\alpha)$ where $f$ is the minimal polynomial of $\alpha$. But then $L = \mathbb F_{p^{sd}} $. Since $\mathbb F_{p^{sd}}$ is unique, we see that this is the splitting field of every irreducible polynomial of degree $d$ over $K$.   Take $K = \mathbb F_2$ and let $P(X) = X^3 + X + 1$, $Q(X) = X^3 + X^2 + 1$. Let $L$ be the splitting field of $P$ and $L&#39;$ be the splitting field of $Q$. The above tells us that $L$ and $L&#39;$ are isomorphic. I would like to construct an explicit isomorphism between $L$ and $L&#39;$.   I know that $L \cong \mathbb F_2[X] /(X^3 +X + 1)$ and $L&#39; \cong \mathbb F_2[X] / (X^3 + X^2 + 1)$. Intuitively, I want to find an isomorphism $\phi : \mathbb F_2[X] \to \mathbb F_2[X]$ such that $\phi((X^3 + X + 1)) = (X^3 + X^2 + 1)$. A little playing around gives me $\phi(X) = X+1$. It now feels like I'm falling at the last hurdle: how do I finish the construction of an isomorphism between $L$ and $L&#39;$? I don't think $\phi$ makes sense as a map from $L$ to $L&#39;$, yet it seems the map I want. AMD-V is not enabled in VirtualBox on AMD APU Binomial confidence interval estimation - why is it not symmetric? I have used the following r code to estimate the confidence intervals of a binomial proportion because I understand that that substitutes for a "power calculation" when designing receiver operating characteristic curve designs looking at detection of diseases in a population.  n is 150, and the disease, we believe, is 25% prevalent in the population. I have calculated the values for 75% sensitivity and 90% specificity (because that's what people seem to do).      binom.test(c(29,9), p=0.75, alternative=c("t"), conf.level=0.95)      binom.test(c(100, 12), p=0.90, alternative=c("t"), conf.level=0.95)   I have also visited this site:    Which is a java page which calculates binomial confidence intervals, and it gives the same answer.  Anyway, after that lengthy set-up, I want to ask why the confidence intervals are not symmetric, e.g. sensitivity is      95 percent confidence interval:    0.5975876 0.8855583      sample estimate probability: 0.7631579    Sorry if this is a basic question, but everywhere I look seems to suggest that they will be symmetric, and a colleague of mine seems to think they will be too. I'm currently working on the following question:  "At least one of Plato and Democritus believed in the theory of forms. Plato believed in the theory of forms only if he was not an atomist, and Democritus was an atomist only if he did not believe in the theory of forms. Democritus was an atomist. Prove, using Natural Deduction, that Plato was not an atomist"  I have been working for quite some time to take the information and turn it into a set of literals and statements I can work with.  So far, I have got $P \lor D$,  $P \Rightarrow \lnot a$,  $D \Rightarrow A$,  $D = A$  However, I cannot seem to make these prove the statement using natural deduction, any hints? scope of clause within complex sentence If I said, &quot;I will run, not if it is snowing, if I my shoes are comfortable.&quot; Does the &quot;if my shoes are comfortable&quot; modify the &quot;I will run&quot; clause? I agree the sentence would be clearer with the addition of &quot;and&quot; in place of the comma. However, without the &quot;and&quot; is it still clear that the &quot;if my shoes are comfortable&quot; clause modifies the &quot;I will run&quot; clause? Why I cannot ping 192.168.0.108 (my computer IP?) How to prompt for user input and read command-line arguments For what $n$ is $U_n$ cyclic? When can we say a multiplicative group of integers modulo $n$, i.e., $U_n$ is  cyclic?   $$U_n=\{a \in\mathbb  Z_n \mid \gcd(a,n)=1 \}$$  I searched the internet but did not get a clear idea. How does centrifugal force work? I know what centrifugal force is, but how does it work? Why are things forced to the outside? How to obtain Hamiltonian formalism and phase space for Lagrangian with second-derivatives? I am trying to get the SIDs of users and groups by querying  get-wmiobject -query 'select groupcomponent,partcomponent from win32_groupuser`   output:  __GENUS          : 2 __CLASS          : Win32_GroupUser __SUPERCLASS     : __DYNASTY        : __RELPATH        : Win32_GroupUser.GroupComponent="\\\\COMPUTER\\root\\cimv2:Win32_Group.Domain=\"TESTDOMAIN\",Name=\                    "Schema Admins\"",PartComponent="\\\\COMPUTER\\root\\cimv2:Win32_UserAccount.Domain=\"TESTDOMAIN\"                    ,Name=\"Administrator\"" __PROPERTY_COUNT : 2 __DERIVATION     : {} __SERVER         : __NAMESPACE      : __PATH           : GroupComponent   : \\COMPUTER\root\cimv2:Win32_Group.Domain="TESTDOMAIN",Name="Schema Admins" PartComponent    : \\COMPUTER\root\cimv2:Win32_UserAccount.Domain="TESTDOMAIN",Name="Administrator"   groupcomponent is a win32_group ().  win32_group is contains an SID ()  However the groupcomponent here is a string and not a win32_group as   get-wmiobject -query 'select groupcomponent.sid from win32_groupuser'   is an invalid query. How can I access groupcomponent's members as a win32_group? Installing Ubuntu after Windows 10 How can I install Ubuntu on Windows 10 as dual boot? I have a DVD with Ubuntu and when I tried to install Ubuntu I had a warning that the hard disk will be erased. How can I install Ubuntu and keep my Windows 10? Thanks What is the principle of underdetermination? While studying I read about the principle of underdetermination of scientific theories. I made some researches online but I am more confused than before. I read about the Quine-Duhem holistic thesis, but I do not know if is the same as underdetermination. Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$ Showing the sequence converges to the square root For any $a &gt; 0$, I have to show the sequence $x_{n+1}$ $=$  $ \frac 12$($x_n+ $ $ \frac {a} {x_n}$)  converges to the square root of $a$ for any $x_1&gt;0$.  If I assume the limit exists ( denoted by $x$) then,  $x$ $=$  $ \frac 12$($x+ $ $ \frac {a} {x}$) can be solved to $x^2 = a$  How could I show that it does exist? How can I remove duplicates in my .bash_history, preserving order? I really enjoying using control+r to recursively search my command history. I've found a few good options I like to use with it: # ignore duplicate commands, ignore commands starting with a space export HISTCONTROL=erasedups:ignorespace  # keep the last 5000 entries export HISTSIZE=5000  # append to the history instead of overwriting (good for multiple connections) shopt -s histappend  The only problem for me is that erasedups only erases sequential duplicates - so that with this string of commands: ls cd ~ ls  The ls command will actually be recorded twice. I've thought about periodically running w/ cron: cat .bash_history | sort | uniq &gt; temp.txt mv temp.txt .bash_history  This would achieve removing the duplicates, but unfortunately the order would not be preserved. If I don't sort the file first I don't believe uniq can work properly. How can I remove duplicates in my .bash_history, preserving order? Extra Credit: Are there any problems with overwriting the .bash_history file via a script? For example, if you remove an apache log file I think you need to send a nohup / reset signal with kill to have it flush it's connection to the file. If that is the case with the .bash_history file, perhaps I could somehow use ps to check and make sure there are no connected sessions before the filtering script is run? Questions about right ideals As i am new to blender i decided to learn by recreating some animations which have some interesting problems, and gain more knowledge about how tv intros are being done, collecting some parody intros to videos. I came across a simple yet troublesome effect , found no answer by googling or getting through tutorials and testing methods on this case led me to nowhere.     I attempted to recreate the animation above, especially  the extending/appearing/drawn white strip with "randomly" placed gaps , strip which travels just above surface , casting shadow.   Tried dynamic paint,( with alpha channel) image sequence texture but although effect was very similar, whole invisible strip was casting shadow being Z transparent, in places where white color haven't reached yet.  I tried bezier curves as well(vector mode) but it appears that taper cant profile part in squary way, or maybe i am missing something here.  Shape keys seem to hard way to go , while keeping whole animation of stripe "unrolling" at constant speed.  I tried as well putting an array of planes on the path but it is completely not what i attempt to achieved.   I will edit post and add some pictures tomorrow. How to safely change OS name in grub boot menu? I have a system where I am dual-booting Ubuntu GNOME 15.10 and Ubuntu GNOME 16.04, however after installing Ubuntu GNOME 16.04, I have found that in the GRUB boot entry list it is called "Ubuntu", whereas the other one is called something like "Ubuntu GNOME 15.10", is there any way I can change the text of the "Ubuntu" one to "Ubuntu GNOME 16.04"? And also change the names of the other options so that instead of "Advanced options for Ubuntu" it says "Advanced options for Ubuntu GNOME 16.04" etc?  I rather do it manually if it is safe to do so, or a piece of software could be recommended, however nothing from a PPA or other untrusted not official source (I will only install software from the Ubuntu official repositories).