Let $\alpha =(a_1,a_2,...,a_s)$ be a cycle and $\pi$ a permutation in $S_n$. Then $\pi\alpha\pi^{-1}$ is the cycle $(\pi(a_1),...,\pi(a_s))$.  I'm having trouble understanding the proof of this:  Take $\pi(a_i)$. Then $\pi\alpha\pi^{-1}\pi(a_i)=\pi\alpha(a_i)=\pi(a_{i+1\; (\mathrm{mod}\;\mathrm{s})})$, then follows on from here.  But I'm not understanding how this proves the statement, since it feels like it's proving that $\pi\alpha=(\pi(a_1),...,\pi(a_s))$.
Prove the following in $S_n$:  Let $\alpha = (a_1,a_2,\ldots,a_s)$ be a cycle and let $\pi$ be a permutation in $S_n$. Then $\pi\alpha\pi^{-1}$ is the cycle $(\pi(a_1),\ldots,\pi(a_s)).$     I'm not sure where to even start.
How can I prepend a tag to the beginning of several files? MacOS High Sierra: An error occurred while verifying firmware How to add a confirm dialog to a command button? I'd think this would work just fine, but it seems to prevent the action function from firing regardless of the confirm response  &lt;apex:commandButton value="Remove All" action="{!removeAll}" rerender="form"      status="LoadingStatus" onclick="return confirm('Are you sure?');"/&gt; How do I create Fortify Alchemy enchantments up to 29%? Prove that   $\dfrac {\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number using number theory. Do not use calculator or Wolfram alpha or anything like that. As I have heard people did not trust Euler when he first discovered the formula (solution of the ) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs.   I believe many of you know some nice proofs of this, can you please share it with us? Trash icon and files have disappeared from the desktop after upgrade to Ubuntu 19.04 How do I add this extra partition to my installed Ubuntu system? I started installing Ubuntu by watching a video tutorial about doing a disk partition installation, for that, inside Windows' 7 own disk partitioning tool. After a defrag I cut 100GB that I pretended to use on my Ubuntu installation for a little try out, now, while installing, I never really chose where I wanted Ubuntu to be installed, so I guess it just made a partition by itself of the minimum that the installation tells you it needs.  How can I now give the other 92 GB I had left apart only for that task?  Here's what my partitions look like now.     Here's a df:    And here's the GParted tool un ubuntu: How to check if value is in range of cells? I want to check if the value of a cell exists in a range of cells. How can I do this?  Just like you would be able to do "if value in list_of_values" in Python for example. I am a Nigerian citizen.  In 2007 I was twice denied a visa to the UK.  Recently I was filling out an application for another UK visitor visa and by mistake I answered no to the question whether I had ever been denied a UK visa, instead of yes.  When reviewing my application, the visa officer found out about the previous visa rejections and as a result I got a 10 year ban on travel to the UK.  What can I do to lift my 10 year ban?   Will this ban affect my chances of obtaining a visitor visa to the USA? Where's the error in this $2=1$ fake proof? I'm reading Spivak's Calculus:  2 What's wrong with the following &quot;proof&quot;? Let $x=y$. Then $$x^2=xy\tag{1}$$ $$x^2-y^2=xy-y^2\tag{2}$$ $$(x+y)(x-y)=y(x-y)\tag{3}$$ $$x+y=y\tag{4}$$ $$2y=y\tag{5}$$ $$2=1\tag{6}$$  I guess the problem  is in $(3)$, it seems he tried to divide both sides by $(x-y)$. The operation would be acceptable in an example such as:  $$12x=12\tag{1}$$ $$\frac{12x}{12}=\frac{12}{12}\tag{2}$$ $$x=1\tag{3}$$  I'm lost at what should be causing this, my naive exploration in the nature of both examples came to the following: In the case of $12x=12$, we have an imbalance: We have $x$ in only one side then operations and dividing both sides by $12$ make sense. Also, In  $\color{red}{12}\color{green}{x}=12$ we have a $\color{red}{coefficient}$ and a $\color{green}{variable}$, the nature of those seems to differ from the nature of $$\color{green}{(x+y)}\color{red}{(x-y)}=y(x-y)$$ It's like: It's okay to do the thing in $12x=12$, but for doing it on $(x+y)(x-y)=y(x-y)$ we need first to simplify $(x+y)(x-y)$ to $x^2-y^2$. A murder of crows? I love the subset of collective nouns known as the terms of venery. These are collective nouns specific to a particular group of animals. Some of the more inventive examples are: a murder of crows, a crash of rhinos, a mischief of mice, and a puddling of ducks (specifically swimming ducks).  Is there a standard for these group names and is there a central reference for them as with other zoological taxonomy?  I can see in the Wikipedia articles that the practice has its origins in Medieval Hunting tradition, but the Kangaroo (Mob), for instance would have been unknown at that time, so how do new collections come into being? Improper integral $\sin(x)/x $ converges absolutely, conditionaly or diverges?  We have that $$\int_1^{\infty}\frac{\sin x}{x}dx$$ $$u=\frac{1}{x}$$ $$du=-\frac{1}{x^2}dx$$ $$dv=\sin xdx$$ $$v=-\cos x$$ $$\int_1^{\infty}\frac{\sin x}{x}dx=-\cos(1)-\int_1^{\infty}\frac{\cos x}{x^2}dx$$ $$\int_1^{\infty}\frac{\cos x}{x^2}dx&lt;\int_1^{\infty}\frac{1}{x^2}dx$$ So it converges. Now I need to find out if $\int_1^{\infty}|\frac{\sin x}{x}|dx$ converges or diverges. Show default image when no uploaded image What does the Recycle Bin in Windows actually do?  Is it just a glorified folder and a holding place for soon to be deleted files, or does it do something specific?  Specifically, are files that are "moved" to the Recycle Bin actually moved on the hard disc, or are just the pointers to the files moved?  I am a fairly experienced user, I just wanted a more depth explanation of the Recycle Bin. Why didn't the rebels on Hoth use X-Wings to fight AT-ATs? do a good job explaining why the Imperial forces didn't have fighter protection for their ground force on Hoth.  But nothing there explains why the Rebels didn't use X-Wings to easily dispatch the AT-ATs and buy more time. They obviously had them available because they were sending them off with transports and some of these were obviously not sent until after "defending" the base (since a few of the pilots of the X-Wings were the Rogue Squadron pilots fighting in the Battle of Hoth).  This happens in the EU in the book  (in fact, the cover picture shows this scene) with great success and Wedge actually comments, on page 120:     Back then we were in airspeeders - undergunned and overmatched... Not the case this time [referring to being in X-Wings against AT-ATs instead of speeders on Hoth]   This indicates X-Wings can perform in cold conditions - there was snow in this environment as described by Corran Horn after he is shot down. The temperature difference between day-time on Hoth and here is unknown however.  Additionally, the X-Wings were able to destroy the AT-ATs with their laser cannons only - no grappling was required.  I suspect the answer is basically "because that would have been boring to watch on-screen" but it seems there may have been an actual in-universe reason outside of this. What is this lock screen with ads and how do I remove it? I think my device got a malware. A lock screen with ads sometimes appear, and I must swipe an area at the bottom of the screen to open the phone. I am in Eastern Europe, some ads are in Slavic languages, and they are seemingly related to travel.    Image resized. Click to view the full image  I have seen ads for Lot Air (Polish carrier). I recently went to their site.  Considering my recent surfing activity and the nature of the ads, I guess the malware is connected to airlines active in Eastern Europe.  The lock screen also promotes the app "DU Quick Charger" by claiming it has recharged the phone extra fast during screen off. (!?) It also shows battery level. I have never installed DU Quick Charger. I doubt DU Quick Charger is installed on the phone.  The problem started after installing Ryanair app. The problem is maybe, but not surely, related to this. Uninstalling Ryanair and restarting the phone doesn't remove the occasional appearance of this lock screen. After a while the situation has changed for the worse, now this lock screen appears after EVERY screen off.  I have always used Avast. I am not rooted and only install via Google Play.  What is this? And how do I remove it? What is the bug about: Keyword gets duplicated even when the query is not modified.  Impact: As far as i can tell - cosmetic.  How to replicate: Go to:  and keep pressing Search button. With each search there will be a duplicate python keyword appended to search box.  Best guess for what is causing it: I think you have somewhere a piece of code that appends tag as a keyword without checking if it is already present. My reasoning comes from this url:  not having the same issue - the tag is not present in it. How to derive variance-covariance matrix of coefficients in linear regression I am reading a book on linear regression and have some trouble understanding the variance-covariance matrix of $\mathbf{b}$:    The diagonal items are easy enough, but the off-diagonal ones are a bit more difficult, what puzzles me is that  $$ \sigma(b_0, b_1) = E(b_0 b_1) - E(b_0)E(b_1) = E(b_0 b_1) - \beta_0 \beta_1 $$  but there is no trace of $\beta_0$ and $\beta_1$ here. Can an irrational number raised to an irrational power be rational?