Find all natural numbers $n$ and $m$ for which $n^m=(n-1)!+1$
To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$
Text not very readable on different image backgrounds I've come here seeking for a bit of design help or advice on how I could possibly improve the following:  Like the title says, I'm having a website banner with multiple text displayed on different images as backgrounds. Now, since most of the images are dark, I've set the font color as white. But even then, some of the text is not very readable on a couple of images like:    Since I'm working with CSS, I've tried different options such as:  Adding Text Shadows:    CSS:  text-shadow: 3px 3px 0px #000;  Also tried, creating a semi-transparent box around the text:      I feel the box one looks out of place.  I've created a  for anyone who knows CSS. If not, please feel free to advice me based on your imagination alone.  PS: You could also advice on using a different font if that would help it stand out better. Could one fire a bullet with sufficient speed to leave the Earth? I just updated my iPhone 7 Plus to iOS 13 tonight and noticed that the post menu in the Stack Exchange app became close to unreadable, because the font/text is too large to be properly displayed. Luckily we can still recognize the icons...    (Yes, app development has been frozen for quite a while but the app continues to serve me well.) How is $\ln(-1) = i\pi$? Let $f \in L^1(\mathbb{R})$, for $a\in \mathbb{R}$ let $f_a(x)=f(x-a)$, prove that: $$\lim_{a\rightarrow 0}\|f_a -f \|_1=0$$  I know that there exists $g\in C(\mathbb{R})$ s.t $\|f-g\|_1 \leq \epsilon$, this is also true for $f_a$ and $g_a$.  Now I have the next estimation:  \begin{align*} \|f_a-f\|_1 &amp;= \int_{\mathbb{R}} |f(x)-f(x-a)| dx \\ &amp;= \int |f(x)-g(x)+g(x)-g(x-a)+g(x-a)-f(x-a)| \\ &amp;\leq \|f-g\|_1 + \|f_a-g_a\|_1 + \int |g(x)-g(x-a)| \end{align*} my question is: I can argue that $\lim_{a\to0} \int |g(x)-g(x-a)| = \lim_{a\to0} \lim_{T\to\infty} |g(x_0)-g(x_0-a)| 2T$, now I think that I can change the order of the limits here, but I am not sure why?  P.s $x_0$ is some point in $[-T,T]$, and the above is valid from the intermediate integral theorem for continuous functions, right?  Thanks. Finder windows just show screen artefacts On our Macbook Air (Mid-2011 11", OS X 10.9.5 Mavericks), starting a few days ago, every time we open Finder windows, the right side just shows screen artifacts/cheese. File Open dialogs in MS Word work fine - it is just the Finder windows with the problem.  Can someone help?  I have tried rebooting, and Verify Disk in Disk Utility. Neither helped. Maybe I should upgrade to El Capitan? But why would this start happening all of a sudden? Show that, if $a$ and $h$ are positive numbers, $h &lt; a^2$, then $$\sqrt{a^2 + h}-a &lt; \frac{h}{2a} &lt; a - \sqrt{a^2 - h}$$  I've been working on this problem for about 2 hours now, but I've made no progress. I'm not looking for an answer, but I just need some help to get me started since we didn't practice inequalities this complex in highschool. Thanks.   All I can tell is that we're supposed to take the square root of an expression at some point since one inequality ( h &lt; a^2 ) becomes two.   Edit : Thank you guys for the replies, but I'd appreciate only hints in the future (like Robert Israel) so that I can learn. Regardless, I found a different way to do it, so it's cool :) Why are processes in Windows 8 32 bit? Let $G$ be a group and $\{H_i\}_{i\in I}$ a family of subgroups. I would like to find a condition that will imply that $\cup_{i\in I} H_i$ is a subgroup. I know that it's not true in general, I need help to find this condition.  Any help is welcome.  Thanks a lot. apt-get update only for a specific repository I know that the Naruto movies are outside the timeline in the sense that they don't affect whatever happens after them. However, they must fall somewhere in the Naruto timeline — based on who are in the movies and how the characters look like, etc. For example, if Sasuke isn't in the movie, it should take place after he left. For each of the movies, approximately what timeframe (between episodes/arcs) are they set?  The movies:   Ninja Clash in the Land of Snow Legend of the Stone of Gelel Guardians of the Crescent Moon Kingdom Naruto Shippuden the Movie Bonds The Will of Fire The Lost Tower Blood Prison Road to Ninja The Last - Naruto the Movie Boruto: Naruto The Movie What does "sudo apt-get update" do? I need an approach to allow delegated admins (i.e., non-Administators) to add members to Public Groups and Collaboration Groups (the GroupMember and CollaborationGroupMember objects).  I added a custom object (UserGroupRequest__c) to capture which Users should be added/removed to which Groups.    I've also built a batchable class to process the requests.  However, GroupMember is a Setup object, and CollaborationGroupMember is not.  So this doesn't work if I combine the operations in a single batchable class.  Should I instead be using 2 batch classes, calling one from the finish method of the other?    Any recommendations on how best to proceed?  Am I way off base?    Would a trigger on the Request object perhaps be better? I'm guessing that there may be a permission problem if this is attempted via a trigger as the delegated admins won't be able to add users to public Groups.  Also, I'm guessing the trigger approach would run into the same mixed DML issue - on Insert of the Custom object, I would need to Insert/Delete from GroupMember and from CollaborationGroupMember, and then perform a 2nd update on the Custom object to indicate that processing was completed... static variables in an inlined function I have a function that is declared and defined in a header file. This is a problem all by itself. When that function is not inlined, every translation unit that uses that header gets a copy of the function, and when they are linked together there are duplicated. I "fixed" that by making the function inline, but I'm afraid that this is a fragile solution because as far as I know, the compiler doesn't guarantee inlining, even when you specify the "inline" keyword. If this is not true, please correct me.  Anyways, the real question is, what happens to static variables inside this function? How many copies do I end up with? Which one is correct: "wish I was here" or "wish I were here"? Possible Duplicate:          Which one is the correct form: "Wish I was here" or "wish I were here"? I've heard both of them many times but I don't know which one is correct. How to remove the Unity Amazon package in 16.04? I was able to go to the ubuntu software center before and remove it and now it doesn't show up since the switch to the gnome software store. Trying the terminal method of removing the shopping lens doesn't work in this release. addplot3[surf] creates lines instead of a surface We have the matrix: $$\left[\begin{array}{rrr} 0.8&amp;0&amp;0\\ 0&amp;1.2&amp;0\\ 0&amp;0&amp;1\\\end{array}\right]$$ Finding the eigenvalues and eigenvectors are easy. But finding ${\bf M}^n$ when $n \rightarrow \infty$ is the hard part. Any suggestions? Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem) 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? I'm looking for a time and memory efficient solution to calculate a  in C. I need to avoid dividing because I'm on a PIC 16 which has no dedicated division unit.  At the moment, I just store all values in a  and simply store and update the sum each time a new value arrives. This is really efficient, but unfortunately uses most of my available memory...