$x^p-a$ either has a root or is irreducible
$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?
Is there a way to write binary literals in C#, like prefixing hexadecimal with 0x? 0b doesn't work.  If not, what is an easy way to do it? Some kind of string conversion? Solve the following system of equations  $ \begin{cases}  x_1^{'}(t)=x_1(t)+3x_2(t) \\  x_2^{'}(t)=3x_1(t)-2x_2(t)-x_3(t) \\  x_3^{'}=-x_2(t)+x_3(t)\end{cases} $.  First, I create the column vectors $X$ and $X^{'}$. Then the matrix $$A= \begin{bmatrix} 1 &amp; 3 &amp; 0  \\ 3 &amp; -2 &amp; -1 \\ 0 &amp; -1 &amp; 1  \\ \end{bmatrix} $$  Now, I find the eigenvalues, $-4,3,1$ and their corresponding eigenvectors $(-3,5,1)^T (-3,-2,1)^T (1,0,3)^T$.   I'm just not sure how to take it from here and solve the system of differential equations. I want a diagonal matrix $D$ so that I can read the solutions easy, but I'm not sure how to do it.  EDIT   Building on @Francisco 's answer, I'd have that:  $$X=c_1  (-3,5,1)^T  e^{-4t}   + c_2 (-3,-2,1)^T e^{3t} + c_3 (1,0,3)^T e^{t} $$.  But I believe this could be written in a simpler form. Merge some tab-delimited files Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $f(f(x) + y) = f(x^2 - y) + 4yf(x)$. Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $$f\big(f(x) + y\big) = f(x^2 - y) + 4yf(x)$$ for all $x,y\in\mathbb{R}$.   $f(x) = 0$ is obvious , but how can we find all other functions?  Thanks in advance! Debian based system, only one gui program, nothing else I would like to make a debian system where you can only open one GUI program at boot, no other graphical interface, no minimize, or any X apart from that program, is there a way to do it on debian if possible, or any other custom distro? I just want to boot open the program and allow the user to only see and use that program. how to get line spacing greater than one and half spacing and less than double space Does anyone know how to get line spacing in a paragraph to be greater than one and half spacing and less than double spacing? I tried \renewcommand{\baselinestretch}{value} but it did not work. The comment pop-up is too far away When you click the add a comment button when you don't have enough rep, you get a pop-up saying that you need more rep in able to comment. Except, it is too far away and users might be confused for a while before realizing the pop-up is on the other side of the page:     Clearly, that is very inconvenient for a lot of people, like me. Can we move the pop-up to be right next to the add a comment button? Evaluation of $\lim_{n\rightarrow \infty}\frac{n!\cdot e^n}{\sqrt{n}n^n}$ without stirling approximation Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\frac{n!\cdot e^n}{\sqrt{n}\cdot n^n}$   $\bf{My\; Try::}$we can write it as $$l=\lim_{n\rightarrow \infty}\frac{e^n}{\sqrt{n}}\cdot \left(\frac{1}{n}\cdot \frac{2}{n}\cdot \frac{3}{n}\cdots \cdots \frac{n}{n}\right)$$  $$\ln (l) = \lim_{n\rightarrow \infty}\bigg[n-\frac{1}{2}n+\sum^{n}_{r=1}\ln\left(\frac{r}{n}\right)\bigg]$$  Now how can i solve it, Help required, Thanks How was Jon Snow able to kill the White Walker in S05E08? The Software Center in Ubuntu 16.04 (Gnome Software) shows only part of the packages available. While e.g. apt list qtcreator correctly lists the corresponding package a search in Software Center finds nothing.  Is there something misconfigured? I upgraded from 15.10. Until now I have used the old Ubuntu Software Center which worked fine. Probability that no two consecutive heads occur? A fair coin is tossed $10$ times. What is the probability that no two   consecutive tosses are heads?   Possibilities are (dont mind the number of terms):  $H TTTTTTH$, $HTHTHTHTHTHTHT$.  But except for those,  let $y(n)$ be the number of sequences that start with $T$  $T _$, there are two options, $T$ and $H$ so,  $y(n) = y(n - 1) + x(n-1) = y(n - 1) + y(n - 2)$  Let $x(n)$ be the number of sequences that start with $H$,   $H _$ the next option is only $T$ hence,  $x(n) = y(n - 1)$   The total number of sequences $F(n)$ is:  $$F(n) = y(n) + x(n) = 2y(n - 1) + y(n - 2)$$  We are after $F(10)$,  $$F(10) = 2y(9) + y(8)$$  $$F(3) = 2y(2) + y(1) = 1 + 4 = 5$$  $$F(4) = 2y(3) + y(2) = 6 + 2 = 8$$  But I'm not quite sure where this will take me though. Numlock turned on on boot-up I have a desktop running ubuntu 10.04  I want numlock to be turned on initially.  What do I have to change to accomplish this?  ~~edit~~  I checked in my BIOS.  It was already set up to have the numlock turned on initially.  Yet it isn't being turned on initially.  I have a USB keyboard: would that break this functionality?   Or, should I ask elsewhere (superuser, perhaps) about how to get that BOIS setting working properly? No Sound (Dummy Output) error Ubuntu 19.10 How to make a good glow effect around a sun? I've installed Process Lasso and in All processes tab, set all affinities to 1-7, meaning that all the processes can be start on all my CPUs except the first one. Yet, there is a huge load on my first CPU.   I personally have no clue what's happening here and really appreciate any help.  My system is:   Intel core i7 - 4720 HQ  2.6 GHz upto 3.6 Windows 10 (64 bit) NVidia GTX 950 M   This screenshot, shows the software's interface. Those 8 green bars on upper right show my CPU usage. Also the column called Rules, indicates the affinities on the processes.       PS: These are screenshots that show the more familiar Task Manager and Resource Monitor. I've sorted all with respect to their CPU usage. Note that the overall CPU usage is just fine (less than 30% most of the time), but in Resource Monitor, same as the Process Lasso, it's obvious that that most of the load is on CPU0 and the others are almost free.  Also I found out that my total CPU usage, is actually more than all my processes altogether! The orange line in Resource Monitor picture, shows the amount of CPU usage of all my processes. (I did the same in safe mode. Orange line, exactly coincides the total usage in safe mode. So I'm suspecting that something is happening in my normal boot.) What is some good free software for doing physics calculations?  I'm mainly interested in symbolic computation (something like Mathematica, but free). What does it do exactly if I click in the window of cmd? Suppose I run cmd.exe and ping localhost. After that I ping localhost again but I click into the window and the ping stops until I press enter.  What happens exactly when I click into the window? Does it suspend the process or something else? I attached a screenshot to clarify this issue.  Note: I do not actually click but drag using the mouse which creates a white box in the window as you can see on the screenshot.  Edit: further clarification: What I really wish to know is whether dragging the window halts the cmd process or not?  Edit: even more clarification: I know that I'm using Quick Edit and pressing enter copies the contents I selected. What I do not know is what happens in the background. Do Legendary Pokemon Respawn After The Elite Four? If $G$ has no proper subgroup, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.     I know that since $G $is a group with no proper subgroups, $g \in G$ is not just the identity. I don't know where to go from there. Short story that takes place off planet, involving sun and rain I read this short story about 20 years ago, and I'd love to find the title and author. This is what I can recall about the story:   It takes place in a classroom full of kids (around 10 years old?) The planet always gets rain The kids lock the main character (a girl, I think) in the closet during the very rare occurrence of sunlight and she misses the entire event.    I know it's not much to go on, but I hope it rings a bell for someone!