Ring/Nilpotent Proof
$1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent
What Nikon lens to buy to make photos of birds? I have a multiboot system set up. The system has three drives. Multiboot is configured with Windows XP, Windows 7, and Ubuntu - all on the first drive. I had a lot of unpartitioned space left on the drive and was reserving it for adding other OSes and for storing files there in the future.  One day I went ahead and downloaded Partition Wizard and created a logical NTFS partition from within Windows 7, still some unpartitioned space left over. Everything worked fine, until I rebooted the computer a few days later.  Now I'm getting:    error: unknown filesystem.   grub rescue&gt;   First of all I was surprised not to find any kind of help command, by trying:    help, ?, man, --help, -h, bash, cmd, etc.  Now I'm stuck with non-bootable system. I have started researching the issue and finding that people usually recommend to boot to a Live CD and fix the issue from there. Is there a way to fix this issue from within grub rescue without the need for Live CD?  UPDATE  By following the steps from , I was able to boot to initramfs prompt. But not anywhere further than that.  So far from reading the manual on , I was able to see my drives and partitions using ls command. For the first hard drive I see the following: (hd0) (hd0,msdos6) (hd0,msdos5) (hd0,msdos2) (hd0,msdos1)  I now know that (hd0,msdos6) contains Linux on it, since ls (hd0,msdos6)/ lists directories. Others will give "error: unknown filesystem."  UPDATE 2  After the following commands I am now getting to the boot menu and can boot into Windows 7 and Ubuntu, but upon reboot I have to repeat these steps.    ls ls (hd0,msdos6)/ set root=(hd0,msdos6) ls / set prefix=(hd0,msdos6)/boot/grub insmod /boot/grub/linux.mod normal   UPDATE 3  Thanks Shashank Singh, with your instructions I have simplified my steps to the following. I have learned from you that I can replace msdos6 with just a 6 and that I can just do insmod normal instead of insmod /boot/grub/linux.mod. Now I just need to figure out how to save this settings from within grub itself, without booting into any OS.  set root=(hd0,6) set prefix=(hd0,6)/boot/grub insmod normal normal   UPDATE 4  Well, it seems like it is a requirement to boot into Linux. After booting into Ubuntu I have performed the following steps described in the :  sudo update-grub sudo grub-install /dev/sda   This did not resolve the issue. I still get the grub rescue prompt. What do I need to do to permanently fix it?  I have also learned that drive numbers as in hd0 need to be translated to drive letters as in /dev/sda for some commands. hd1 would be sdb, hd2 would be sdc, and so on. Partitions listed in grub as (hd0,msdos6) would be translated to /dev/sda6.  UPDATE 5  I could not figure out why the following did not fix grub:  sudo update-grub sudo grub-install /dev/sda   So I downloaded  based on an answer from  post. That seemed to do the trick after I picked the "Recommended Repair (repairs most frequent problems)" option. If tenured staff are virtually unsackable, why is the drive to find funding so strong? Why are disabled apps still running? proving that elements of a sequence are pairwise coprime For the sequence $a_n = 6^{2^n} + 1$:   How can I prove that the elements of this sequence are pairwise coprime, i.e. prove that if $m$ is not equal to $n$ then $\gcd(a_m, a_n) = 1$. (begin by proving that $a_n∣ (a_{n+1} - 2)$) How do I create a hinged door that opens and closes in Blender? If I knew how to do this, then it would help me also create moving control surfaces on model aircraft, such as the rudder and horizontal stabilizers. On a whim the other day, I changed my meta name from Lord Torgamus to Popular Demand.  I just noticed that the old name has "stuck" to the author tag of ; everywhere else on meta, my new name displays as expected.  Is this a feature?  Is it a bug?  Mostly I'm just curious, as I can't think of a strong reason in favor of keeping this behavior or a strong reason for changing it.  A question somewhat like this has been raised before as , but the previous poster implied that the issue had something to do with migration, which my situation shows is not the case. How to know that your machine learning problem is hopeless? How do you remove a default program association for file types in Windows 7? I somehow have set the default file type association for a file type in Windows 7. It doesn't appear to be possible to delete it from the "Change the file type associated with a file extension" options screen. Is it possible to remove it? How can I customize the Ubuntu installer? How do you find time period as a function of time for a simple pendulum that is in the form of a hollow sphere that is filled with mercury and there is a hole in the bottom through which the mercury is constantly falling at a fixed rate? I tried creating a function for the time period to check how it varies with the mass of mercury, but I had too many variables.   EDIT: This question was a concept based multiple choice question in my homework book. The option were time period becomes erratic, time period increases, time period decreases, time period first increases then decreases. What is the module/method used to get the current time? How to specify ec2 instance types for different services? After I have specified a default EC2 instance type for my Juju charm, how do I override that for one of the services?  For instance, the  deploys wordpress and mysql, and I can set the default-instance-type in my environments.yaml so both services are m1.small  But how do I configure it so that (only) the mysql service gets deployed on an instance type better suited to a database, like m2.xlarge ? nameref (hyperref) evaluating counter instead of section name What's the proper way to reference a section with a counter in the title? The TOC generates correctly, but the cross references aren't so lucky.  MWE:  \documentclass[letterpaper, 12pt, twoside]{exam}  \usepackage{hyperref}  \newcounter{KSRLessonNumberCounter} \setcounter{KSRLessonNumberCounter}{1} \newcommand{\KSRSection}[2][\value{KSRLessonNumberCounter}]{         \setcounter{KSRLessonNumberCounter}{#1}         \section*{Lesson \arabic{KSRLessonNumberCounter}: #2}         \addcontentsline{toc}{section}{Lesson \arabic{KSRLessonNumberCounter}: {#2}}         \stepcounter{KSRLessonNumberCounter} } \newcommand{\KSRSubsection}[1]{         \subsection*{#1}         \addcontentsline{toc}{subsection}{#1} } \newcommand{\KSRLessonNumber}{ }  \begin{document}  \KSRSection{Sample}  This is the first lesson. The second lesson is \nameref{sec:blah}.  \KSRSection{Blah} \label{sec:blah}  This is the second section  \KSRSection{Lorem Ipsum}  Dollars are Set in \nameref{sec:blah}.  \end{document} Rear fender angle For a 20" rear wheel with a this kind of fender, what would be the best angle to install it to minimize the spray on one's butt/legs? Uniform convergence of derivatives, Tao 14.2.7. User deleting code after question has been answered If a user keeps  (thus making the answer meaningless), what should we do? Rollback the edit (what if the user just deletes the code again)? Delete the question?  More examples (same user): ,  (see also edit histories). Using $g = \frac{Gm}{r^2}$, the force on a point mass located at 1 AU from the Sun ($m = 2 \cdot 10^{30} \text{ kg}$) is about ~0.006 N/kg.  Does that mean that, e.g., a 70 kg person is ~42g lighter during the day, and ~42g heavier at night? That seems like it could make a big difference for things like measuring gold bars or other weight-sensitive items. (Gold arbitrage: buy your gold during the day and sell it during the night! Risk-free profit!)  This makes me suspect that I'm overlooking something obvious, because a ~0.05% weight difference seems like something everyone would have noticed long ago. So what am I missing?    Edit: A few answers below indicate that there shouldn't be any weight difference, because the Earth orbits the Sun in free-fall. But if that's the reason, does that mean that a 1:1 tidally-locked Earth-Sun system wouldn't experience differential gravity from the Sun on opposite sides? That doesn't seem right. Logistic Regression - Adding interactions makes Independent variable statistically insignificant My name is Abhi &amp; I am trying to better understand logistic regression by solving a few practice problem. I am using R and RStudio as the development environment  Problem Statement  Given the age, sex and class(first,second,third) for each passenger can you predict if he survived or died when the titanic sank   Attempt1  Simple Logistic Regression. Use age,sex and passenger class directly. The formula (in R)  Survived~Pclass+Sex+Age  This gives fairly decent results - accuracy of 79% and all of independent variables are statistically significant   Attempt2  Add interactions between age,sex and passenger class. Accuracy has improved to 80% but age is no longer significant. Also none of the new terms(age-sex,age-class,sex-class,age-sex-class) are statistically significant. The formula (in R) Survived~Pclass*Sex*Age   Can some one explain why this is happening? I can accept that the new terms may not be significant but why is age no longer significant?   Any help would be much appreciated Facebook message notification showing blank message Since I updated the system to 10.11.5, the Facebook message notification is not showing message text.   Is this a bug in this software upgrade of OSX?