Addition and multiplication are symmetric and associative. But I have no idea about operators which are symmetric but not associative. Please help me listing any such 2-arity operators.
Does commutativity imply associativity? I'm asking this because I was trying to think of structures that are commutative but non-associative but couldn't come up with any. Are there any such examples?  NOTE: I wasn't sure how to tag this so feel free to retag it.
Conky does not work properly on Ubuntu 16.04 GNOME I have this strange problem with conky. It doesn't show properly on the desktop:    I've tried everything from editing conky widget file to reinstalling conky and conky manager (v.2.4). Who knows how to fix it? In Avengers: Age of Ultron movie,      Ultron infected tech of Avengers Tower just after his birth. It means he is capable of infecting tech. Also, if this wasn't the case, he could never be able to create his drones.   Just after its birth,      He escaped using the Internet and infected digital devices across the entire planet. Again, awesome capability (think Skynet).   We also saw that      Iron Man's armor was connected to the Internet. He googled for the keyword "Hulk" during flight just before he deployed Veronica.   Then, why did Ultron let Iron Man win the fight? Why didn't he simply hack Iron Man's armor to use him as own drone? Step to reproduce  Answer a question; Answer gets accepted or upvoted; Receive notification in the app; Change the title of the question; Notification is duplicated with updated title.  Expected behavior When going into notifcations you should see only one entry for that reputation event. Actual behavior You get two notification entries. In the screen notice the two entries for the same question, the one related to EF. After answering I have changed the name to make it more searchable for future questions and i've noticed it.      App Version: 1.6.1.2 Device: iPhone 6s OS Version: Version 10.1.1 (Build 14B100) Is there an Autocorrect feature in TeXStudio 2.6.2? I was wondering if there's an Autocorrect feature in TexStudio 2.6.2 that works like in MS Word so that if I have to type a certain word many times, the editor writes it for me as soon as I press spacebar after I finish typing the shortcut for that particular word. (Eg. phl --> photolithography)   I've already searched StackExchange for similar questions and read Qrrbrbirlbel's comment  . I've tried my hand at the Macros but it's not working properly, maybe some bug that needs TeXStudio to re-start for it take action or maybe I simply don't know how to make it work.  Can someone help me, please? I think sometimes there is a need to post question as anonymous, reason being to avoid linking content of a question with the identity of the questioner. This may not be so evident or required on technical sites like stackoverflow.com but sometimes is a necessity on sites like  or maybe in future depression.stackexchange.com  On such sites user may not want to link his identity to some questions where he may detail about his being over-weight or skinny or maybe some other attribute which he may like to keep private and avoid making his concerns public, but still want to ask a question with valid data.  I am not suggesting that user asking question should not be registered or not responsible for content. I think all down-votes or up-votes should still be attributed to user, but this way most users would not suffer , and reputation could even be used for a bounty. The only change should be that while asking question a valid registered user can hide his identity (the whole user signature block/user card and the reputation should be replaced by "identity-hidden") and that question should not be listed in his public profile.  Anyway identity of questioner doesn't add much value to question. If we think such anonymous question may reduce social nature of Stack Exchange sites, and maybe majority of questions will become anonymous we can curb that by putting a daily or monthly limit per user or put a price of say 50 reputation per anonymous question.  PS: I don't think this is duplicate of        because motive for question and reason for feature request are different, and I propose a different solution too, because creating another account for asking such questions has many problems which having a single account doesn't have    I don't have to manage other account If I keep single account I still may want to keep my details to be linked to my non-anonymous questions I may want to use reputations I earned as bounty for anonymous questions    also allows anonymous questions and may be we can take some hint from there. Proof of $\lim_{n\to \infty} \sqrt[n]{n}=1$ Thomson et al. provide a proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$ in . It has to do with using an inequality that relies on the binomial theorem:  I have an alternative proof that I know (from elsewhere) as follows.  Proof. \begin{align} \lim_{n\rightarrow \infty} \frac{ \log n}{n} = 0 \end{align} Then using this, I can instead prove: \begin{align} \lim_{n\rightarrow \infty} \sqrt[n]{n} &amp;= \lim_{n\rightarrow \infty} \exp{\frac{ \log n}{n}} \newline &amp; = \exp{0} \newline &amp; = 1 \end{align}  On the one hand, it seems like a valid proof to me. On the other hand, I know I should be careful with infinite sequences. The step I'm most unsure of is: \begin{align} \lim_{n\rightarrow \infty} \sqrt[n]{n} = \lim_{n\rightarrow \infty} \exp{\frac{ \log n}{n}} \end{align} I know such an identity would hold for bounded $n$ but I'm not sure I can use this identity when $n\rightarrow \infty$. Question: If I am correct, then would there be any cases where I would be wrong? Specifically, given any sequence $x_n$, can I always assume: \begin{align} \lim_{n\rightarrow \infty} x_n = \lim_{n\rightarrow \infty} \exp(\log x_n) \end{align} Or are there sequences that invalidate that identity?  (Edited to expand the last question) given any sequence $x_n$, can I always assume: \begin{align} \lim_{n\rightarrow \infty} x_n &amp;=  \exp(\log \lim_{n\rightarrow \infty} x_n) \newline &amp;=  \exp(\lim_{n\rightarrow \infty} \log x_n) \newline &amp;=  \lim_{n\rightarrow \infty} \exp( \log x_n) \end{align} Or are there sequences that invalidate any of the above identities? (Edited to repurpose this question). Please also feel free to add different proofs of $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$. What are the rules about using an underscore in a C++ identifier? It's common in C++ to name member variables with some kind of prefix to denote the fact that they're member variables, rather than local variables or parameters. If you've come from an MFC background, you'll probably use m_foo. I've also seen myFoo occasionally.  C# (or possibly just .NET) seems to recommend using just an underscore, as in _foo. Is this allowed by the C++ standard? I do love to listen to music while watching text YouTube videos, but I can't because YouTube can't be muted.  I use the respect audio focus in  Music Player.  I have tried App Volume Control but it did nothing! Printing pointers in C I really get confused when to use say and when to use tell. Which is appropriate in the following:   What did he tell? / What did he say? What are you saying? / What are you telling? Can I use a Regex in an XPath expression? Something like .//div[@id='foo\d+] to capture div tags with id='foo123'.  I'm using .NET, if that matters. How can I convert byte size into a human-readable format in Java? Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism. Allow sound file uploads in questions and answers? Should Stack Exchange introduce the ability to upload embedded sounds (maybe via HTML5/Flash, whichever the user has) as part of questions and answers to Stack Exchange?  It could require a certain amount of rep to stop new users from doing it frivolously.  My rationale behind this is the new language &amp; usage sites.  I often find myself thinking that it might be helpful, when discussing a regional dialect or something, for the person answering to just pronounce what they're trying to describe in words. Proof that a sequence is convergent I'm asked to prove the convergence of the sequence $$X_n=\left(1+\frac12\right)\left(1+\frac14\right)\left(1+\frac18\right)\cdots\left(1+\frac{1}{2^n}\right)$$   I proved that it is increasing through the ratio test and then I want to prove that it is bounded.  My question is the following : is it enough to say that $X_n&lt;\left(\frac32\right)^n$ for $n&gt;1$ or not ? Namely, can we say that a sequence is bounded if $\forall$  $n$ ,  $\exists$ a $N$ such that $X_n&lt;N$ or do I have to find a N such that $\forall$  $n$ , $X_n&lt;N$?  In other words, can a bound depend on $n$ ?  Thanks for your help. I'm sure there is a really simple explanation to this but haven't found any pointers on this forum or elsewhere. I seem to have lost the 'processing' menu from QGIS and can't figure out how/why!? Any ideas what I might have done to make this happen?  I'm running QGIS 2.8 built against GDAL 1.10 on Ubuntu 14.04. Density of random variables Let $X$ and $Y$ be independent random variables with respective densities: $f_X (x) = xe^{-\frac{x^2}{ 2}} 1_{]0,+\infty[} (x)$ and $ f_Y (y) =  \frac{1}{\pi \sqrt{1-y^2}} 1_{]-1,1[} (y)$  Let $U:=XY$ and $V:=X\sqrt{1-Y^2}$  I try to check whether U and V are independent and which law they follow.  What I did:  After some calculations, I found the density of $(U,V)$:  $ f_{(U,V)}(u,v)=\frac{1}{\pi } e^{-\frac{u^2}{ 2}}1_{\mathbb{R}}e^{-\frac{v^2}{ 2}}1_{\mathbb{R}_+^* }$ Which mean that $(U,V)$ follows a $ \mathcal{N}(0,1)$ law (because $u^2+v^2=x^2$) and that  U and V are independent because $f_{(U,V)}$ can be factored as the products of densities of $U$ and $V$  My question:  There is a factor $\frac{1}{\pi}$ and I don't know to which density it will belong: to $U$ density or to $V$ density?  Many thanks! Using Sequence to delete a part of an expression I got a positive definite matrix $B,$ that is, $V(x) = x^TBx &gt; 0$ for any vector $ x \neq 0.$  I want to show that $ \lambda_\min \|x\|_2^2 \leq V(x) \leq \lambda_\max \|x\|_2^2$ for any $x \neq 0,$ where $\lambda_\min$ and $ \lambda_\max$ are defined by $$ \lambda_\min = \min \lbrace | \lambda|: \lambda \text{ is an eigenvalue of } B \rbrace$$ and  $$ \lambda_\max = \max \lbrace | \lambda|: \lambda \text{ is an eigenvalue of } B  \rbrace$$  Any hint please? Can I have too many mods? I found a lot of mods that make thinks lighter, and add mark and recall, and a basement I like. They are all on Steam, and what I want to know is if I can have too many? Will they all work together? Or should I just leave them alone?