How to fix GRUB's "file not found." error
After update, get "error: file not found." followed by "grub rescue> _"
I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.  Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate $x$ with coefficients in $R$. We say that a polynomial $f = \sum\limits_{i=0}^n r_ix^i \in R[x]$ (with coefficients $r_0, r_1, \ldots, r_n$) is primitive if $\langle r_0,r_1,...,r_n\rangle = R$, i.e., the ideal generated by the coefficients of $f$ is $R$. Let $f$ and $g$ be two polynomials in $R[x]$. Prove that $fg$ is primitive iff $f$ and $g$ are both primitive.  The $\Rightarrow$ part is easy. Say, $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$; then $fg = \sum\limits_{i=0}^{m+n} c_ix^i$, where $c_k = \sum\limits_{i + j = k}r_is_j$. Since $fg$ is primitive, there exists a set of $\{\alpha_i\} \subset R$, such that $\sum\limits_{i=0}^{m+n} \alpha_ic_i = 1$, to prove $f$ is primitive, I just need to write all $c_i$'s in terms of $r_i$'s, and $s_j$'s, then group all $r_i$ accordingly, rearranging  it a little bit, and everything is perfectly done. And the proof of the primitivity of $g$ is basically the same. The $\Leftarrow$ part is just so difficult. Say $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$ are both primitive, then there exists $\{\alpha_i\}; \{\beta_i\} \subset R$, such that $\sum\limits_{i=0}^{n} \alpha_ir_i = 1$, and $\sum\limits_{i=0}^{m} \beta_is_i = 1$. At first, I thought of multiplying the two together, but it just didn't work. So, I'm stuck since I cannot see any way other than multiplying the two sums together. I hope you guys can give me a small push on this. Thanks very much in advance, And have a good day. What should be maximum and minimum length of persons User Name & Password? I am wondering because I am trying to make sure that person have right amount of spaces for their user name.  User Name and Password are security measures and should not be easy to guess. I have fixed amount of characters to Passwords no shorter than 6 characters because most sites use 6-8 characters when create an account. Because of limitation of my progams database file I cannot have passwords longer than 20 characters. What about User Name field? Too short User Name would be too easy to guess, too long, might cause program errors (since maximum length of single line TextBox in .Net/C# is about 32767 characters).  What should be the maximum and minimum length of User Name and Password fields? Dual booting Windows and Ubuntu My laptop has 500gb hard disk in which for 250gb I installed Windows xp on C drive(primary partition). I want to install Ubuntu 12.04 in the remaing hard drive with dual boot option. When I am going to install Ubuntu it is showing something like  /boot /boot/   Can anybody help me in this regard that where and how to install Ubuntu? Horizontal line inside a matrix I tried to make a horizontal line inside a matrix by the attached code. However, it didnot work the compiling process gives a time out message. It work normally without the \hline command. Any help!  \documentclass[journal,draftcls,onecolumn,12pt,twoside]{IEEEtran} \usepackage{array} \usepackage{blkarray} \begin{document} \begin{equation}  \centering W_b(1)=\begin{blockarray}{cccccc} \begin{block}{(ccccc)c}  a      &amp; b     &amp;     &amp;  &amp;  &amp; \matindex{\leftarrow d=0} \\ \ddots &amp; \ddots&amp;     &amp;  &amp;  &amp; \matindex{\vdots} \\    &amp;     a &amp; b   &amp;  &amp;  &amp; \matindex{\ \ \ \   \leftarrow d=d_b-1} \\   %\hline  c      &amp; d     &amp;     &amp;  &amp;  &amp; \matindex{\leftarrow d=d_b} \\   \ddots &amp;\ddots &amp;     &amp;  &amp;  &amp; \matindex{\vdots} \\    &amp;   c   &amp;  d  &amp;  &amp;  &amp; \matindex{\leftarrow d=D}  \end{block} \end{blockarray}, \label{eq:eq4}  \end{equation}  \end{document} Due to this question and answer , I assume 1k = 1000, 2k= 2000...  I don't believe it. Or maybe I am wrong. Please check the below images.        You will see 2kviews. But when you navigate to it, the actual view count is 15xx.         What is going on? Is it a bug? Alternatives to one-way ANOVA for heteroskedastic data I have data from 3 groups of algae biomass ($A$, $B$, $C$) which contain unequal sample sizes ($n_A=15$, $n_B=13$, $n_C=12$) and I would like compare if these groups are from the same population.  One-way ANOVA would definitely be the way to go, however upon conducting normality tests on my data, heteroskedascity seems to the main issue. My raw data, without any transformation, produced a  ratio of variances ($F_{\max} = 19.1$) which is very much higher than the critical value ($F_{\rm crit} = 4.16$) and therefore I cannot perform one-way ANOVA.   I also tried transformation to normalise my data. Even after trials of various transformations (log, square root, square), the lowest $F_{\max}$ produced after transformation with a $\log_{10}$ transformation was $7.16$, which was still higher compared to $F_{\rm crit}$.  Can anyone here advise me on where to go from here? I can't think of other methods of transformation to normalise by data. Are there any alternatives to a one-way ANOVA?  P.S.: my raw data are below:   A: 0.178 0.195 0.225 0.294 0.315 0.341 0.36  0.363 0.371 0.398 0.407 0.409 0.432     0.494 0.719 B: 0.11  0.111 0.204 0.416 0.417 0.441 0.492 0.965 1.113 1.19  1.233 1.505 1.897 C: 0.106 0.114 0.143 0.435 0.448 0.51  0.576 0.588 0.608 0.64  0.658 0.788 0.958 why is the output of the following code coming to be zero? Conservation Of Angular Momentum of a falling stone I am currently stuck on the following question.   A stone is dropped from a stationary helicopter 500m above the ground, at the equator. How far from the point vertically below the helicopter does it land and in what direction? Solve this problem by conserving angular momentum.   My current interpretation is that when the stone falls in order to converse angular momentum the angular velocity must increase. This leads to the stone travelling to the east.   I equated the angular momentum at the start       L0=*(re+h)mVo  where re is radius of earth, h is the height above that, m is mass and Vo is inital velocity. To the angular momentum at any time  L=(re+h-1/2gt^2)m(Ve+Vo)  where the 1/2at^2 term comes from gravitational acceleration and Ve is the additional velocity.   I rearranged to find Ve and then integrated over the time it takes to fall the 500m in order to work out the extra distance the stone traveled. However, my answer is 1/2 what the answer should be. I have checked the maths and it seems to be fine. So I believe my set up is wrong. Any help would be appreciated. Standalone python app that uses qgis.core without installing qgis Are the "umbles" etymologically related? From      And we show telltale signs of early hypothermia, what the doctor and   the National Institutes of Health and other cold-weather cognescenti   call the "umbles."      First we lose some fine motor skills, McLaren says, "and we start to   fumble. And tumble. And stumble."      Then our cognitive abilities are compromised. "We mumble and we   grumble."   Are the "umbles", such as fumble, tumble, stumble, mumble and grumble, etymologically linked, either in terms of sharing a common ending, or entire words having a common ancestor?  Trying to do a search of the etymology of the umbles ended up giving the etymology of the word  instead. Aliens want to destroy all dogs because they are seen as next step in evolution beyond Man I read a SF short story years ago, I think it may have been in high school back in the 1970s. Aliens come to earth, to kill all the dogs, because they see Man as destroying himself, and dogs as evolving to take Man's place, thus being a threat to the aliens. Humanity agrees to kill all the dogs to get rid of the aliens, but in fact have taken border collies (?) deep underground in order to save them and ensure that they evolve.  Does anyone know the title/author of this short story?  Thank you in advance for your assistance. Google chrome is reading characters backwards What's the proof that the Euler totient function is multiplicative? That is,     why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers?   It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most. And why the minimum value for $\varphi (n)$ should be $ \frac{4n}{15} $ completely passes me by. Wrong numeration in toc longer then one page A word for a person who always needs there to be a problem? Is it ever bad to use \left and \right? For mathematics, is it ever disadvantageous to use \left(  and \right)  in equations?  I can't see any way that doing this would hinder the typesetting, so why isn't () just converted into \left( \right) automatically? This would save a lot of typing.  Cheers! How to copy files via terminal? If my question gets migrated to another site, is it possible somehow to reverse it back to the original site? I have a txt file which actually is a html source of some webpage. Inside that txt file there are various strings preceded by a "title=" tag.  e.g.   &lt;div id='UWTDivDomains_5_6_2_2'  title='Connectivity Framework'&gt;    I am interested in getting the text Connectivity Framework to be extraced and written to a separate file.  Like this, there are many such tags each having a different text after the title='some text here which i need to extract '  I want to extract all such instances of the text from the html source/txt file and write to a separate txt file. The text can contain lower case, upper case letters and number only. The length of each text string(in characters) will vary.  I am using PowerGrep for windows. Powergrep allows me to search a text file with regular expression inout. I tried using the search as        title='[a-zA-Z0-9]  It shows the correct matches, but it matches only first character of the string and writes only the first character of the text string matched to the second txt file, not all string.   I want all string to be matched and written to the second file.  What is the correct regular expression or way to do what i want to do, using powergrep?  -AD. Question, is it ok for one of the pages levels to 404? (SEO-wise)  Let's say I have a sitemap like this:  example.com --/services   --/foo   --/bar --/sectors   --/my-sector   --/another-sector   --/the-sector   --/stuff --/about-example --/contact --/etc --/etc   This makes a lot of structural sense. However, from a design point of view, I think having a page to show all the sectors (example.com/sectors) is pointless since all of the sector pages can be accessed from the home page, main navigation, and in each sector page.  Would there be any SEO consequences if the /sectors page were to 404?  Note: in the sitemap 'example.com/sectors' would not be included.