Extracting specific range of characters from each line
Is there a command like head but for line truncation?
If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative Let $R$ be a ring such that for every $x\in R$ we have $x^2-x \in Z(G)$. Show that $R$ is a commutative ring.    My thoughts  What should I do? I could show that every $y \in R$ could be written in the shape $x^2-x$ but I don't know if that is true. We know that  $y \cdot(x^2-x) \ = \ (x^2-x)\cdot y$ for any $x,y \in R$, but that doesn't mean that $xy = yx$. Please give me a hint bordermatrix with blocks I need to write a complicated matrix, that has a structure as in the attached picture. Why did Voldemort hide two Horcruxes in Hogwarts? How are connections differentiated from behind NAT routers? Does separability imply the Lindelöf property? Does separability imply a sort of Lindelöf property?  Since I can't prove this fact I'm beginning to think that my conjecture is false.  Intuitively, $\mathbb{R}$ has a countable subset $\mathbb{Q}$ which is used to form a countable basis for $\mathbb{R}$ with the usual topology and prove the Lindeöf property. Solve differential equation, $x'=x^2-2t^{-2}$ Solve differential equation:  $x'=x^2- \frac{2}{t^2}$  Maybe is it sth connected with homogeneous equation? I have no idea how to solve it. I understand the reasoning behind the fairly narrow textwidth in a default LaTeX article. 60-80 characters per line and all that. However, I am preparing a document that features many lengthy equations, and it seems silly to me to restrict the equations to fit into the same margins as the text when there's all that whitespace going unused on either side.  I'd like to have equations which are wider than the textwidth do the obvious thing, that is, continue to be centered on the page just like all the other equations, and simply extend into both the left and right margin equally. My equations are unnumbered, so there's no concern about where to put equation numbers.  How might I go about implementing this? What is the /boot partition really for? I'm reading a relatively old text on Linux partitions and file systems (the LPIC 1 Certification Bible). It says:     Some versions of the Linux boot loaders cannot access a kernel that is   outside the first 1024 cylinders on a disk. By putting the /boot   partition at the beginning of the drive you can be assured of not   having a problem when accessing the kernel at boot. This problem shows   itself most often in cases of dual booting Linux along with another   operating system that is on the first partition.   Why would a boot loader have "no access to the kernel outside the first 1024 cylinders on a disk"?  Also, what does "putting the /boot partition at the beginning of the drive" mean? Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$  Then how prove $G$ is an abelian group?  Some context: Some of these commutation relations often imply that $G$ is abelian, for example , or . This looks like another example of this phenomenon, but the same techniques do not apply. For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , show $V= \ker T^n \oplus $ Im $T^n$ Suppose that X and Y are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the combined estimator of $\theta$ $$ \tilde{\theta} = k_1 X + k_2 Y $$ where $k_1$ and $k_2$ are arbitrary constants.   Find the condition on $k_1$ and $k_2$ such that $\tilde{\theta}$ is an unbiased estimator of $\theta$. For $\tilde{\theta}$ unbiased, show that the variance of the estimator is minimized by taking $k_1 = 1/3$ and $k_2 = 1/3$. Given observations $x$ and $y$ find the maximum likelihood estimate of $\theta$ and hence show that $\tilde{\theta}$ is also the maximum likelihood estimator.   I have gotten (1) and (2) okay, but it's (3) I am having trouble with, I'd be okay if $X$ and $Y$ had the same parameter but I'm having trouble with $X$ and $Y$ having different parameters, any help would be appreciated.   NOTE   For (1) I got $k_1 = 1 - 2k_2$.   For (2) I found the variance of $\tilde{\theta}$, then differentiated and let equal to zero to minimize - therefore we get (after subbing in $k_2 = 1 - k_1/2 $) $$3k_1-1=0,$$ which when subbing in $1/3$, we see it is minimised.   Thank you. Local file access with JavaScript How do you do load testing and capacity planning for web sites? This is a  about capacity planning for web sites. Related:      What are some recommended tools and methods of capacity planning for web sites and web-applications? Please feel free to describe different tools and techniques for different web-servers, frameworks, etc., as well as best-practices that apply to web servers in general. Induction with floor, ceiling $n\le2^k\implies a_n\le3\cdot k2^k+4\cdot2^k-1$ for $a_n=a_{\lfloor\frac{n}2\rfloor}+a_{\lceil\frac{n}2\rceil}+3n+1$ This was posted on facebook and people are saying it is incorrect, it should be: "...as you and I"   Which is correct? Is there a --parents option for mv like for cp? cp has a useful feature --parents that allows a file's directory structure to be copied to another directory:  $ cp -v --parents source/directory/from target/ ‘source/directory/from’ -&gt; ‘target/source/directory/from’   How do I similarly move a file? If $f$ is continuous on $[a,b]$, $f(x)≥0$ on $[a,b]$ and $$\int_{a}^{b} f(x) =0$$   then prove that $f(x)=0$ for all $x \in [a,b]$.  I tried with Riemann's definite integral definition but couldn't proceed Is char *envp[] as a third argument to main() portable If statement evaluating true no matter what. Python 3.3.0 I know how much space is required to install Ubuntu.  How much space does a Ubuntu installation take up on the disk?