Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$.
In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
Center of a free product Let $A$ and $B$ be two groups and let $G=A*B$ be their free product. Suppose both $A$ and $B$ are nontrivial. I want now to prove that $G$ is infinite and that $Z(G)$ is trivial.  I think I can prove that $G$ is infinite using this argument: two reduced words of $A*B$ are equal if and only if they have the same length and the ith letter of the first word is the same as the ith letter of the second one. Hence one can find a reduced word of $G$ of length arbitrarily big (since we are sure it won't be equal to a shorter word). Do you think it works? Has anyone another suggestion?  On the other hand, I don't really have a clue for the second part except trying to analyze a long list of cases of reduced words and prove for each one that the only element with which they commute is the identity. Does anyone have a smarter idea?  Thanks in advance! Style/class tutorials What are some good tutorials on writing custom styles (classes) that clearly explain commands used and some of the features that TeX has to offer in the way of styles? Show that $u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$ is in $W^{1,n}(U)$, where $U=B(0,1)\subset\mathbb{R}^n$. The entire problem statement is:  Let $n&gt;1$ and let $U=B(0,1)\subset\mathbb{R}^n$. Show that $u:U\to\mathbb{R}$ given by  $$u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$$ is in $W^{1,n}(U).$  My attempt at the proof is as follows:  To show that $u\in W^{1,n}(U)$ it suffices to show that $|Du|\in L^{n}(U)$. Consider $$u_{x_i}=\frac{1}{\ln\left(1+\frac{1}{|x|}\right)}\frac{1}{1+\frac{1}{|x|}}|x|^{-2}\frac{x_i}{|x|},$$ which simplifies to, $$u_{x_i}=\frac{1}{\ln\left(1+\frac{1}{|x|}\right)}\frac{1}{1+\frac{1}{|x|}}\frac{x_i}{|x|^3}.$$ Thus, $$|Du|=\frac{1}{\ln\left(1+\frac{1}{|x|}\right)}\frac{1}{1+\frac{1}{|x|}}\frac{1}{|x|^2}$$ which can be manipulated to, $$|Du|=\frac{1}{\ln\left(1+\frac{1}{|x|}\right)}\frac{1}{1+|x|}\frac{1}{|x|}.$$  Moreover, since $U=B(0,1)$ we can bound the second term from above which gives, $$|Du|\leq\frac{1}{\ln\left(1+\frac{1}{|x|}\right)}\frac{1}{|x|}$$  Thus, we can instead show that $$\int_U\left(\frac{1}{\ln\left(1+\frac{1}{|x|}\right)}\frac{1}{|x|}\right)^n&lt;\infty.$$  Converting this into polar coordinates we have with $r=|x|$, $$\int_0^R\int_{\partial B(0,r)}\left(\frac{1}{\ln\left(1+\frac{1}{r}\right)}\frac{1}{r}\right)^n\,dSdr=\int_0^R\left(\frac{1}{\ln\left(1+\frac{1}{r}\right)}\frac{1}{r}\right)^n\int_{\partial B(0,r)}dSdr$$ Note that $\int_{\partial B(0,r)}\,dS=n\alpha(n)r^{n-1}$, which is the surface area of $B(0,r)$. Thus, we have then $$n\alpha(n)\int_0^R\left(\frac{1}{\ln\left(1+\frac{1}{r}\right)}\frac{1}{r}\right)^nr^{n-1}\,dr=n\alpha(n)\int_0^R\left(\frac{1}{\ln\left(1+\frac{1}{r}\right)}\right)^n\frac{1}{r}\,dr$$  And so this is where I get stuck in the problem, since I don't how I can evaluate that last integral to show it's finite.  Thank you for any help or feedback! Yes, I'm a moron, and I tried to manually install a higher point release of python 2.7.16 to /usr/bin/  (i did ./configure --prefix=/usr) on my Ubuntu 16.04 machine. It seems to work fine EXCEPT for packages that depended on python, like apt-listchanges, which fails with messages like ImportError: No module named apt_pkg  there's no make uninstall available, I've tried manually removing files, and still when I try apt --reinstall install python, Nothing gets overwritten and the python executable still gives the new 2.7.16 version...   Any idea how to get the official repository version back?? How to get back sudo on Ubuntu? The usage of singular and plural has always been confusing for me.   I often see sentences like these         People are using cell phones.   People are using a cell phone.      Does the first sentence mean everyone has a phone and they are all using their own?  Does the second sentence mean they are sharing one cell phone?   If I see a group of people holding a cell phone in their hand(s), &lt;- even this is confusing for me, should I use the first sentence then?   Another example: you see two men, and both of them are carrying a bag. Which sentence should I say/use?        They are carrying backpacks          They are carrying a backpack      Could you please make it clear for me? Insect identification - Is this a bedbug? This bug was found on a couch in . The couch, and the two beds in the room show no signs of other bugs. Is this a bed bug that I should worry about or something harmless? It was slightly larger than a common tick. Xcode 6.1 won't let me create a new project I'm running Xcode 6.1 on OS X Yosemite and I'm unable to create a new project, as you can see below it just says 'loading'.  I've removed the product and org name for the purpose of this image. I have tried uninstalling but that didn't work. I am new to Xcode and coding in general so I would appreciate it if any solution could be given in lay terms. How are echo and print different in PHP? Possible Duplicate:          Is there any major and fundamental difference between these two functions in PHP? I have a (symmetric) matrix M that represents the distance between each pair of nodes. For example,       A   B   C   D   E   F   G   H   I   J   K   L A   0  20  20  20  40  60  60  60 100 120 120 120 B  20   0  20  20  60  80  80  80 120 140 140 140 C  20  20   0  20  60  80  80  80 120 140 140 140 D  20  20  20   0  60  80  80  80 120 140 140 140 E  40  60  60  60   0  20  20  20  60  80  80  80 F  60  80  80  80  20   0  20  20  40  60  60  60 G  60  80  80  80  20  20   0  20  60  80  80  80 H  60  80  80  80  20  20  20   0  60  80  80  80 I 100 120 120 120  60  40  60  60   0  20  20  20 J 120 140 140 140  80  60  80  80  20   0  20  20 K 120 140 140 140  80  60  80  80  20  20   0  20 L 120 140 140 140  80  60  80  80  20  20  20   0   Is there any method to extract clusters from M (if needed, the number of clusters can be fixed), such that each cluster contains nodes with small distances between them. In the example, the clusters would be (A, B, C, D), (E, F, G, H) and (I, J, K, L).  I've already tried UPGMA and k-means but the resulting clusters are very bad.  The distances are the average steps a random walker would take to go from node A to node B (!= A) and go back to node A. It's guaranteed that M^1/2 is a metric. To run k-means, I don't use the centroid. I define the distance between node n cluster c as the average distance between n and all nodes in c.  Thanks a lot :) I was wondering why the software center doesn't download all the requested apps and install one by one as they finish to download. Or instead why the software center doesn't download an app while the other is installing How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $? If it is possible to express  $ \cos n \alpha $ in terms of $ \cos \alpha $  as a power series for integer $n$ ...  I like to see an expression for the quotient angle that obviously tallies when $ (\alpha , \beta) $ are swapped.  EDIT1:  Something like:  $$ \cos (\alpha - \beta)= \cos \alpha \cos \beta  + \sin \alpha  \sin \beta $$  EDIT2:  Like to know why $ \cos ( \alpha / \beta )$ cannot be expressed in terms of $ \cos \alpha, \cos \beta $, but  $ \cos ( \alpha + \beta) $  can be expressed in terms of  $\cos \alpha $ and $ \cos \beta. $ With a two-party system, our nation will divide the people (who/whom)   most need to be brought together.   Do I use who or whom for this sentence?  I think that "people" is the direct object and warrants the use of "whom", but I want to make sure I'm right. Two definitions of $\limsup$ Here are two equivalent definitions of $\limsup_{n\rightarrow\infty} a_n$:   Let $u_n=\sup\{a_n, a_{n+1}, a_{n+2},\ldots\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} u_n =    \lim_{n\rightarrow\infty}\left(\sup\{a_n,a_{n+1},\ldots\}\right)$$ Let $E$ be the set of all subsequential limits of $\{a_n\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \sup E$$   I'm curious as to which one people usually learn first, or which one people find more intuitive. Baby Rudin only has the second definition, but doesn't mention the first as far as I know.  As far as intuition, I think of the second definition as "the biggest limit point." I'm not sure how to think of the first definition though. I can see that $u_n \ge u_{n+1}$ for all $n$, but I'm not sure how to interpret the limit of those $u_n$ conceptually.  Are there any other ways to conceptualize $\limsup$? OK guys I have this problem:  For $x,y,p,q&gt;0$ and $ \frac {1} {p} + \frac {1}{q}=1 $ prove that $ xy \leq\frac{x^p}{p} + \frac{y^q}{q}$  It says I should use Jensen's inequality, but I can't figure out how to apply it in this case. Any ideas about the solution? What does "Incompatible types: void cannot be converted to ..." mean? Floating point comparison Why does chmod 777 -R / leave the system unusable? I'm only granting permission to all to do anything but why does the system crash by giving permissions only? I'm only modifying the permission not changing the files. OS: Ubuntu 12.04  Accidentally deleted dpkg. Now, packages can't be installed. So, needs an dpkg installer which installs packages. How to download packages from a command-line given repository?