Is there any algorithm that produces irreducible polynomials over $\mathbb{F}_2$?
irreducibility of a polynomial over $\mathbb{F}_2$
Values of the limit $\lim\limits_{x\to+\infty}\left(\sqrt{(x+a)(x+b)}-x\right)$ Hi I have a question regarding finding the values of limit for the following question.  Let $a, b \in \mathbb R$. Find the limit $$\lim_{x\to+\infty}\left(\sqrt{(x+a)(x+b)}-x\right)$$ scheduled event What is another word for 'all round' capabilities? I am looking for a positive, passionate word to describe the all-round capabilities of a person.   Are there any words better than 'all-rounder'? After the events on Miller's planet, it seems to come as a revelation to Brand that     Miller had only landed a couple hours ago, and died a few minutes ago.   The crew of the Endurance, even when they were back at NASA headquarters, had all the information they needed to figure out the time dilation and how long Miller had been on the planet. As such, they must have known that     Miller, even in the best case, would have only had a couple hours worth of data.   Why use all those resources (fuel and time) just for that? How do I login as root? What can I do to increase battery life on my Android device? What can I do to get better battery life on my Droid? Battery life seems awful. Less than 15 hours or so on normal usage. Proving that free modules are flat (without appealing projective modules) Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first proving that every free module is projective, and then proving that every projective module is flat. Unfortunately, Atiyah &amp; Macdonald's "Introduction to Commutative Algebra" (Chapter 2) does not discuss projective modules. But the result that every free module is flat comes very handy in the exercises.   So my question is,     Is it possible to prove that every free module is flat just by   definitions and without appealing to projective modules?   Thanks! JavaScript get clipboard data on paste event (Cross browser) Removing Katoolin from Ubuntu 14.04 Transposing rows and columns I have a file with the lines like below.  title1:A1 title2:A2 title3:A3 title4:A4 title5:A5  title1:B1 title2:B2 title3:B3 title4:B4 title5:B5  title1:C1 title2:C2 title3:C3 title4:C4 title5:C5  title1:D1 title2:D2 title3:D3 title4:D4 title5:D5  How can I achieve this?   title1    title2     title3    title4 A1         A2         A3         A4 B1         B2         B3         B4 C1         C2         C3         C4 D1         D2         D3         D4 Prove  that the function $\displaystyle f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$. You should give a proof that is directly based on the  definition of continuity.   I saw many similar questions on this website, but none of them gave me a satisfactory answer. I more or less understand how epsilon-delta proofs work but my problem lies more in the algebraic manipulations probably. So here's my "attempt".  Definition of continuity; for $f:A \to \mathbb{R}$     $$\forall \varepsilon &gt;0 \ \exists\delta&gt;0\ \forall x \forall x_0 \in A:|x-x_0|&lt;\delta \Rightarrow |f(x)-f(x_0)|&lt;\varepsilon$$   So from what I understand, the procedure for epsilon-delta proofs are as follows: 1. Plug the point of interest $x_0$ into $|f(x)-f(x_0)|&lt;\varepsilon$ 2. Write $x$ in terms of $\varepsilon$ 3. Then plug $x$ into $|x-x_0|&lt;\delta$ 4. Find $\delta$ in terms of $\varepsilon$. Done    At step 1, we have $\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Step 2 is where I'm stuck because of the 2nd degree polynomial in the denominator. First, I did this: $\left|\dfrac{x}{x^2+1}\right|&lt;\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Then, I wasn't too sure about what to do, I thought about getting rid of the "$1$", but that would screw up my inequality. I thought about factoring $x^2+1$, but it doesn't have any real roots. I tried inverting the fraction, but that led me to nowhere. And I've already exhausted the tools I have. How do I proceed?  Apparently, epsilon-delta proofs are the most basic building blocks of analysis. And I'm already failing, this is both embarrassing and depressing... I appreciate the mechanisms which help bring attention to new answers to old questions - front page sorting, and the moderation tool explicitly designed for it. But I wonder if we can do more. I'm particularly interested in addressing answers whose validity has changed. In the worst case, they were correct when they were posted, but no longer are, so they mislead all the current users who find their way to them looking for an answer. More benignly, perhaps the best answer was good, but slightly flawed.  At this point, a helpful user may post a new answer, which is much better than existing upvoted and accepted answers. It can be difficult to overcome that momentum. What if the default sorting method took this into account? A naive suggestion: if a new answer has a large positive vote velocity, and an old answer has a negative vote velocity, perhaps the default sorting should place the new answer above the old answer. This could help make new answers visible to users who need a correct answer, and also help garner more votes to increase the confidence of the sorting decision.  I know this would be a fairly low-impact change, but nothing is permanent, and I'm afraid of answers with 100 upvotes being pinned at the top for years after they've become obsolete.  Edit: I would welcome other proposals that accomplish the same goal - my naive suggestion wasn't intended to be a complete solution, just one thing that a better sorting method would do. I am trying to create an spritz app. Everything was working fine, but since yesterday I keep getting this error:     ./spritz: line 176: syntax error: unexpected end of file   I have checked the script file and everything seems perfect. I am confused, I have an if statement at last and it looks correct! Here is the last portion:  #checks if speed is 150 157 if [[ $2 -eq 150 ]]; 158 then 159 starttime=$SECONDS 160      FS=$'\n' 161      for j in `grep --color=always -iP '\b[^aeiou\s]*[aeiou][^aeiou\s]*\K[aeiou]' $1`; 162      do 163            #Reads the text file in the centre of the screen 164            echo "                                                    ___________________" 165            echo "                                                             $j"; 166            echo "                                                    ___________________" 167            echo "                                                                               Speed 150 wpm" 168            sleep  0.9; 169            clear; 170       done 171 endtime=$(($SECONDS - $starttime)) 172            echo "You read $words_read words in $endtime seconds!" 173       exit 8 174 fi   What could cause that error? What does $b_i\mid b_{i+1}$ mean in this context? In the computational topology literature, the reduction algorithm for computing the Smith normal form of a boundary matrix uses the notation $b_j &gt; 1 \: \text{ and }\: b_j\mid b_{j+1}$ in the context of the diagonal elements of the Smith matrix. Can anyone give me an idea for what it means? NSDictionary with ordered keys I have an NSDictionary (stored in a plist) that I'm basically using as an associative array (strings as keys and values).  I want to use the array of keys as part of my application, but I'd like them to be in a specific order (not really an order that I can write an algorithm to sort them into).  I could always store a separate array of the keys, but that seems kind of kludgey because I'd always have to update the keys of the dictionary as well as the values of the array, and make sure they always correspond.  Currently I just use [myDictionary allKeys], but obviously this returns them in an arbitrary, non-guaranteed order.  Is there a data structure in Objective-C that I'm missing?  Does anyone have any suggestions on how to more elegantly do this? Can't install any app, App Store asks for credit card I haven't been installing anything on my iPad Mini for the past few weeks. But previously, I just had to tap on "Install", enter my Apple ID password and go.  Now, for every new application, even those free ones, I have to confirm each time, that I'm 17 or over. And then App Store asks to enter my credit card information. If I disagree and hit "Cancel", installation process is interrupted and I can't install any application, even free one.  I don't trust Apple at all and I'm not willing to share my credit card information, especially that I'm installing only free applications.  I'm pretty sure that I haven't changed anything in the configuration of my iPad Mini for ages.   Am I missing some significant change to the App Store or this is some kind of bug or virus? Is there anything I can do in this situation? How can I install a free application from the App Store, without having to share my credit card information with Apple? How can I obtain an 'unbalanced' grid of ggplots? I need to find a period-3 orbit for a tent map defined by $$T(x)=\begin{cases}2x &amp; 0\le x\le0.5 \\[2ex] 2-2x &amp;0.5 &lt; x \le 1 \end{cases}$$  So, what is a fast and efficient way of finding it? I only know how to do it by guessing-and-checking. For example, I picked a random point, $x = \frac{2}{7}$ and found that $$T\left(\frac{2}{7}\right)=\frac{4}{7}\\[4ex]T\left(\frac{4}{7}\right)=\frac{6}{7}\\[4ex]T\left(\frac{6}{7}\right)=\frac{2}{7}$$ which is a period-3 orbit. Is there any better way of finding a period-3 orbit , more specifically, a period-3 point? Is it common to review papers assigned to your supervisor? Is it common to supervisors to email their students papers to review (which were apparently assigned to the supervisor, and not to the student). The field is Computer Science, but I  would also like to hear from the other fields. As you may know, Facebook allows users to choose a "vanity URL," allowing easy linking to one's profile. However, once set, this URL cannot be changed and this was done for security reasons, understandably.  I have a friend whose now ex-partner set the URL to be "herfirstname.hislastname" without her permission. She tried to change it after this, but to no avail.  Has anyone successfully contacted Facebook and got them to either disable the name and revert back to an ID or change the name to another?