I am aware this question has been asked many times but I'm still struggling to understand a few things.  Let R be a ring in which $x^2=x$ for all x ε R.  (a) Prove that $x + x = 0$ for for all x ε R.  (b) Prove that R is commutative.  For (a):   I understand that $a+a=(a+a)^2=a^2+a^2+a^2+a^2=a+a+a+a$. But I don't understand how $a+a+a+a ⟹ a+a=0$.  For (b): For any $x, y$ in the ring $R$, $x+y=(x+y)^2=x^2+xy+yx+y^2=x+y+xy+yx,$ so $xy+yx=0$. But how do I get to $xy = yx$ ?
Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?  Attempt:  by the definition of characteristic of a ring, the first statement implies that characteristic of the ring is $\leq 2$. If the ring is assumed to be non trivial ring then, characteristic is 2.  About the last statement can we construct a field of say 8 elements and show that there exists a an element of order $\neq 2$?  Is there a simpler example ?
Cannot install R-forge package using install.packages Only ftps with proftpd on debian 'A / One / At least one student entered the room.' Are these the same? (truth-conditionally) I just wonder if the two following sentences are truth-conditionally the same. Sentence 1 essentially means there existed a student who entered the room, and this situation includes two, three or more students entering the room. Hence, I believe these two sentences are truth-conditionally the same. Are they?        A student entered the room.   At least one student entered the room.      How about        One student entered the room      I gather this is different, as sentence 3 only means one student, not more and not fewer?  Any and all insights would be welcome. How to efficiently partition a single Windows-Ubuntu dual boot disk? This is not a question on  or  but is a question on access rights and how not to let them bite you in the lower abdomen...  ;-) This question will not help you how to make partitions on your disk, but will tell you what to do in , depending on your personal use case, so choose one of the following:      Note: The above questions can also be used with multiple disks, but the examples in the answers are given for one disk. How do I code a table to look like this? I'm not the best at LaTeX and can't seem to wrap my head around \multirow. I checked other examples prior to posting but they used \multicolumn and it just confused me even more. Thanks in advance!    \begin{center} \begin{tabular}{ |c|c|c|c|c|}   \hline  TextA &amp; TextB &amp; TextC &amp; TextD &amp; TextE\\  \hline  A &amp; TextF &amp; TextG &amp; TextH &amp; TextI\\  B &amp; TextJ &amp; TextK &amp; TextL &amp; TextM\\  C &amp; \multirow{2}*{TextF} &amp; TextG &amp; TextH &amp; TextI\\  \cline{3-3}  &amp; TextJ &amp; TextK \\ \hline \end{tabular} \end{center}   I was trying something like this.. I'm starting to have problems as I'm trying to multirow within a multirow? I'm not sure of the process here. I would like TextG to be split into 2 and TextH and TextI to be the cells which are split in 3 but not sure how to go about it. Will then TextJ and TextK be the contents of the split TextG cell? What regulations should a European driver know when driving in the US? There are the obvious ones. The US has the keep-your-lane system, where as in Europe you are expected to keep to the right (left for UK) as long as possible.  On my latest travel to the US, I was confronted with the possibility to go through a red light if you turn right, unless it is explicitly not allowed.  What should a european driver know when driving in the US? What are the do's and don'ts compared to driving in Europe. When does Ubuntu 16.04 use /etc/apache2/ssl/apache.crt? I have some Ubuntu 16.04 server running and set up one vhost in apache2 which uses (or should use) ssl.  &lt;IfModule mod_ssl.c&gt;         &lt;VirtualHost *:443&gt;          ServerAdmin myemail@mailhoster.xx         ServerName my.domain.name.de         DocumentRoot /var/www/mysslsite          &lt;Directory /var/www/mysslsite&gt;                 Options FollowSymLinks MultiViews                 AllowOverride All                 Order allow,deny                 allow from all         &lt;/Directory&gt;          ErrorLog ${APACHE_LOG_DIR}/mysite.ssl.error.log         CustomLog ${APACHE_LOG_DIR}/mysite.ssl.access.log combined          &lt;/VirtualHost&gt; &lt;/IfModule&gt;   I did NOT enable the standard sites-available/default-ssl.conf.  I am able now to browse to  and it is asking in firefox to add some exception which is totally normal if i did not buy any certificate.  But I was wondering where it is set up that it should use the certificates in /etc/apache2/ssl/ directory. I can not find any config where it is told to use that. I all the time was thinking that it will use something from /etc/ssl folder.   Or did I maybe not activate SSL correctly? Graphical interface for generating presentation slides Is there a good graphical interface, similar to PowerPoint or LibreOffice Impress, for generating presentation slides?  When writing documents I prefer to write my text and LaTeX-markups using a text-editor and not using a graphical tool like LyX. However, in the case of presentation slides I want to see the layout and to have the ability to immediately see what you get e.g. by moving a picture with arrow keys.  ADDITION: The background to my question is as follows. I have used PowerPoint for more than a decade and like to use a WYSIWYG tool when making presentation slides. However, I can not use PowerPoint on my current computer and I think the output from LibreOffice Impress is inferior. As a long-time user of LaTeX, I have considered using SliTeX or Beamer but unfortunately I do not like the way of working. The end result is of course superior, but I found it very time-consuming and tiresome to e.g. adjust the position and size of pictures. Is soaking beans 24 hours unrefrigerated safe? Homomorphisms from $\mathbb{Z} \rightarrow S_3$. I'm trying to compute this integral, $$\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx \hbox{ where } 0&lt;a&lt;1$$ I drew a typical Pacman contour with branch cut at positive real axis. Then, we have $$\int_{\Gamma}\frac{\ln z}{z^{a}(z+1)}dz=\left(\int_{L_{1}}+\int_{L_{2}}+\int_{C_{R}}+\int_{C_{\epsilon}}\right)\frac{\ln z}{z^{a}(z+1)}dz$$ The function has a pole at $z=-1$. Then, by residue theorem, the whole integral becomes  \begin{align*} \int_{\Gamma}\frac{\ln z}{z^{a}(z+1)}dz&amp;=2\pi i\text{res}(\frac{\ln z}{z^{a}(z+1)};-1)\\&amp;=2\pi i\lim_{z\rightarrow-1}(z+1)\frac{\ln z}{z^{a}(z+1)}\\&amp;=2\pi i\lim_{z\rightarrow-1}\frac{\ln z}{z^{a}}\\&amp;=2\pi i\frac{\ln e^{i\pi}}{e^{i\pi a}}\\&amp;=2\pi i\frac{\ln1+i\pi}{e^{i\pi a}}\\&amp;=-\frac{2\pi^{2}}{(e^{i\pi})^{a}} \end{align*} On $L_1$, $z=xe^{i\epsilon}$ where $\epsilon\leq x\leq R$  On $L_{2}$, $z=xe^{i(2\pi-\epsilon)}$ where $\epsilon\leq x\leq R$  On $C_{R}$, $z=Re^{i\theta}$ where $\epsilon\leq\theta\leq2\pi-\epsilon$  Lastly on On $C_{\epsilon}$, $z=\epsilon e^{i\theta}$ where $\epsilon\leq\theta\leq2\pi-\epsilon$  So I tried to compute them separately and I anticipate the curves will both go to $0$ by ML inequality, and two linear integrals will consist of the term $\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx$ and hence simply solve for it. But I'm having trouble physically solving it out. I think the log is throwing me off. Can anybody please help me? I have a 3TB Seagate Backup Plus Desktop USB 3.0 drive, which works fine when in its enclosure, but when I get it off its enclosure and directly plug it as an internal SATA drive, it's just not properly recognized (it works again when used in the enclosure). My systems’s motherboard is an , which has two SATA 6.0 Gb/s and four SATA 3.0 Gb/s ports.  When used as an internal SATA drive Windows 7 asks if I want to format the drive, as if it didn’t have a proper filesystem, and if I use the DISKMGMT.MSC tool (typing that in the start menu) I get completely wrong information about the drive.  It says the drive has 3 partitions (349,31GB of RAW data and two unassigned partitions of 1698,68GB and 746,52GB). This information is plain wrong, since the drive, when used in the USB enclosure, works as a single NTFS partition (2794,52GB). How come it’s shown as 3 partitions without filesystem when connected through SATA?!  Is Seagate using some proprietary way of storing the data when using the drive in its provided USB 3.0 enclosure? I didn’t use any kind of encryption. I just plugged the drive with USB 3.0 and started using it straight away, since it apparently came pre-formatted with an NTFS filesystem. How to get the keyboard to input Japanese? Undefined control sequence error on \fancy@reset, \f@ncyolh and \f@ncyorh when using newlfm class Role of dependencies while installing software?? What is the role of dependencies while installing any software?I was installing vlc player and it were closed by giving me error something like check dependency .  What is to solution of this and how i'll get these dependencies??  Thanks in Advance. Chat flag notification, but no flag It appears  is back. Upon refreshing any chat page (even when dumping the cache and hard reloading), a chat flag notification is displayed, but there is no chat flag. The name of the boy vs.The boy's name Which one is correct or are both correct?     The boy's name is nice.   The name of the boy is nice.     I'm really confused these two sentences?  Which one is more suitable? It seems that containing lava in solid glass, stone and iron blocks may not be sufficient to prevent wool from burning, even if the lava is completely encased.         After some delay, it seems that fire will jump onto the wool above the iron.   Are solid blocks more effective than airspace, or not? If so, Does it matter whether the corner block is filled in? What are the exact rules on distance that I need, and how can I prevent flammable blocks from being eliminated by nearby lava? Bonus: Does a flowing lava block behave differently than a lava source block when it comes to fire creation? Is there a way I can see a list of all my YouTube comments over time? I posted something the other day and wish I could find the video again. Is there a way I can see all of my comments? How to use Noto Color Emoji with lualatex? I want to use Noto Color Emoji with lualatex (package fonts-noto-color-emoji on my box). However, this fails as follows. How to use Noto Color Emoji with lualatex?  \documentclass{article} \usepackage{fontspec} \setmainfont{Noto Color Emoji} \begin{document} 😺 👅  \end{document}   Error:  $ lualatex emoji.tex  This is LuaTeX, Version 1.07.0 (TeX Live 2019/dev/Debian)   restricted system commands enabled. (./emoji.tex LaTeX2e &lt;2018-12-01&gt;  luaotfload | main : initialization completed in 2.224 seconds (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo)) (/usr/share/texlive/texmf-dist/tex/latex/fontspec/fontspec.sty (/usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty (/usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty (/usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3-code.tex) (/usr/share/texlive/texmf-dist/tex/latex/l3kernel/l3pdfmode.def))) (/usr/share/texlive/texmf-dist/tex/latex/fontspec/fontspec-luatex.sty (/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty (/usr/share/texlive/texmf-dist/tex/latex/base/tuenc.def)) (/usr/share/texlive/texmf-dist/tex/latex/fontspec/fontspec.cfg))) (./emoji.aux) [1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] (./emoji.aux))  409 words of node memory still in use:    3 hlist, 1 vlist, 1 rule, 2 glue, 3 kern, 1 glyph, 5 attribute, 48 glue_spec , 5 attribute_list, 1 write nodes    avail lists: 2:15,3:4,4:1,5:21,6:1,7:13,8:1,9:6 &lt;/usr/share/fonts/truetype/noto/NotoColorEmoji.ttf ! error:  (file /usr/share/fonts/truetype/noto/NotoColorEmoji.ttf) (ttf): sfnt  table not found !  ==&gt; Fatal error occurred, no output PDF file produced!