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How to remove time from date? Today I was thinking about composition of functions.  It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.  But then I thought about commutativity.  My first intuition was that bijective self maps of a space should commute but then I saw some counter-examples. The symmetric group is only abelian if $n \le 2$ so clearly there need to be more restrictions on functions than bijectivity for them to commute.  The only examples I could think of were boring things like multiplying by a constant or maximal tori of groups like $O(n)$ (maybe less boring).  My question: In a euclidean space, what are (edit) some nice characterizations of sets of functions that commute? What about in a more general space?   Bonus: Is this notion of commutativity important anywhere in analysis? If a stock doesn't pay dividends, then why is the stock worth anything? I read somewhere that companies are not required to pay dividends to shareholders (this is correct, yes?).   If so, then if company A never pays dividends to its shareholders, then what is the point of owning company A's stock? Surely the right to one vote for company A's Board can't be that valuable.   What is it that I'm missing? Ubuntu 20.04 run ok on Lenovo ThinkPad E15 laptop? Detecting unused packages in LaTeX document I have quite a bad habit of copy-and-pasting my packages from one LaTeX document to another. Is there any way we can detect unused packages (i.e. \usepackage commands) so that they can be removed from the LaTeX document? right now   I have setup /etc/fstab like this:  //192.168.5.167/H /mnt/ssd cifs credentials=/root/.smbreds,noauto,x- systemd.automount 0 0   I want to set it up like first network should be connected and then fstab mount should get mounted. Assigning Float Pointers in C When is $2^n -7$ a perfect square? This came up while solving another ENT problem. I want to ask when is: $$2^n -7 \text{ where } n\geq 3$$ a perfect square? Specifically, I also wanted to know what would be the solutions when $n$ is odd? How should I solve this?  I can check that $n=3, 4, 7$ are solutions but cannot find more. As it is of the form $4k+1$, it doesn't help as well. Develop rich API exposing Stack Overflow functionality I would love to see an actual native iPhone app for Stack Overflow, Server Fault, Super User, etc.  I've actually developed a number of iPhone apps myself, and would certainly be interested and willing in doing the bulk of the legwork, but it would be necessary for the SO team to expose a real API, instead of simply RSS feeds.  Maybe if there's enough demand behind this, we can convince the SO team to build a rich API, which could be useful for many other applications, not just an iPhone app. Analysing Bounce Rate and Time on Page I'm working on an eCommerce site and there's a perceived problem with the bounce rates on product pages. The best performing page (in terms of views), tends to have a very high bounce rate, at about 80%. This page is also targeted with Ads, but in both non-paid and paid traffic, the bounce rate stays high.  What's weird about it is that the time on this page is about 2 and a half minutes. The others pages have a slightly lower bounce rate (between 50% and 75%) with visit times between 1 minute and 2 minutes.  I'm not entirely sure about how to analyse this data. I've checked all the possible technical problems (faulty GA code, JS race conditions, breaks in the chain), and whilst there were some things that need improving, no technical fix budged the bounce rate at all.  The other thing we noticed was that the product pages were getting lots of traffic for very generic and brand based search terms. In these cases the pages come up as sitelinks, could this explain a high bounce rate too?  So in short :  If the page view time is so high, should I be worrying about the bounce rate at all? What's the chances of someone visiting the page and then leaving to compare prices? And how do sitelinks affect bounce rate on eCommerce sites? Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$ How would you go about proving that $$\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$$  for any two integers $a$ and $b$?  Intuitively it is true because when you divide $a$ and $b$ by $\gcd(a,b)$ you cancel out any common factors between them resulting in them becoming coprime. However, how would you prove this rigorously and mathematically? Ubuntu 17.10 windows' menu in panel I installed the latest version of Ubuntu (17.10) , but the problem is, I can't find the menu bars of applications in the GNOME panel. For instance, the Chrome menu bar doesn't appear on the panel like in Ubuntu 17.04 and earlier versions. I would like to configure Apache 2 running on Kubuntu to execute Perl CGI scripts. I've tried some steps that I came across by googling, but nothing seems to work.  What is the right way of achieving this? Union of subgroups is a subgroup if and only if one subgroup is a subset of the other Let $H$ and $K$ denote two subgroups of a group $G$.  Prove that the union $H \cup K$ is a subgroup of $G$ if and only if $H \subset K$ or $K \subset H$. Prove that if $A$ is an infinite set then $A \times 2$ is equipotent to $A$ I want to prove that if $A$ in an infinite set, then the cartesian product of $A$ with 2 (the set whose only elements are 0 and 1) is equipotent to $A$.  I'm allowed to use Zorn's Lemma, but I can't use anything about cardinal numbers or cardinal arithmetic (since we haven't sotten to that topic in the course).  I read a proof of the fact that if $a$ is an infinite cardinal number, then $a+a=a$, which is something similar to what I want to prove.   Any suggestions will be appreciated :) How should we handle downvote retaliation? Show that SO(n) is a normal subgroup of O(n).  A normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. SO(n) is the set of orthogonal matrices of determinant 1. O(n) is the set of real matrices whose inverses equal their transposes (orthogonal matrices). I'm simply bad at writing proofs. Please help? "If" and "Whether" - Is interrogative IF always replacable by WHETHER? Is it worth it to buy an FD-EOS adapter, or is it better to exchange a misbought lens? I recently bought a manual-focus (since manual doesn't bother me) . However, I didn't notice at the time of purchase that it was an FD lens. (I'm a hobbyist; I didn't know Canon had actually changed lens mounts at some point.)  On the one hand, I kind of want to experiment with this guy. And quality FD lenses are relatively cheap on the used market, so if the adapters were worth it, I could probably get more (and more varied) glass. On the other hand, everything I read about the FD-EOS adapters is highly contradictory. As in, some people swear by them, and others think they're outright tripe.  I am still within the exchange period, and can easily exchange this lens for this  that I'd been tossing up about buying anyway, as they are the same price.  Which would be the wiser choice? Can't find boot-repair package for the newest version of Ubuntu