I have seen a [picture like this] several times:    featuring a "troll proof" that $\pi=4$. Obviously the construction does not yield a circle, starting from a square, but how to rigorously and demonstratively prove it?  For reference, we start with a circle inscribed in a square with side length 1. A step consists of reflecting each corner of figure $F_i$ so that it lies precisely on the circle and yielding figure $F_{i+1}$. $F_0$ is the square with side length 1. After infinitely many steps we have a figure $F_\infty$. Prove that it isn't a circle.  Possible ways of thinking:   Since the perimeter of figure $F_i$ indeed does not change during a step, it is invariant. Since it does not equal the perimeter of the circle, $\pi\neq4$, it cannot be a circle.   While it seems to work, I do not find this proof demonstrative enough - it does not show why $F_\infty$ which looks very much like a circle to us, is not one.   Consider one corner of the square $F_0$. Let $t$ be a coordinate along the edge of this corner, $0 \leq t \leq 1$ and $t=0, t=1$ being the points of tangency for this corner of $F_0$ and the circle. By construction, all points $t \in A=\{ \frac{n}{2^m} | (n,m\in \mathbb{N}) \&amp; (n&lt;2^m)\}$ of $F_\infty$ lie on the circle. I think it can be shown that the rest of the points, $\bar{A}=[0;1] \backslash A$, lie in an $\varepsilon$-neighbourhood $U$ of the circle. I also think that in the limit $\varepsilon \to 0$, points $ t\in\bar{A}$ also lie on the circle. Am I wrong in thinking this? Can we get a contradiction from this line of thought?   Any other elucidating proofs and thoughts are also welcomed, of course.
What is wrong with this proof?    Is $\pi=4?$
I exported my QGIS project with the qgis2web plugin as OL3.  Can I embed this folder to a wordpress homepage or can I import it to google maps and add this to WordPress? I am downloading macOS Sierra from Mac App Store. But I have electricity problem here. Due to which my iMac will lose power after 2 hours. Now I want to pause the downloading so that I can resume the download later on. How can I do that?  Here is a screenshot. I am getting the following error trying to read from a socket. I'm doing a readInt() on that InputStream, and I am getting this error. Perusing the documentation this suggests that the client part of the connection closed the connection. In this scenario, I am the server.  I have access to the client log files and it is not closing the connection, and in fact its log files suggest I am closing the connection. So does anybody have an idea why this is happening? What else to check for? Does this arise when there are local resources that are perhaps reaching thresholds?    I do note that I have the following line:  socket.setSoTimeout(10000);   just prior to the readInt(). There is a reason for this (long story), but just curious, are there circumstances under which this might lead to the indicated error? I have the server running in my IDE, and I happened to leave my IDE stuck on a breakpoint, and I then noticed the exact same errors begin appearing in my own logs in my IDE.  Anyway, just mentioning it, hopefully not a red herring. :-( When a 'blur' event occurs, how can I find out which element focus went *to*? Suppose I attach an blur function to an HTML input box like this:  &lt;input id="myInput" onblur="function() { ... }"&gt;&lt;/input&gt;   Is there a way to get the ID of the element which caused the blur event to fire (the element which was clicked) inside the function? How?  For example, suppose I have a span like this:  &lt;span id="mySpan"&gt;Hello World&lt;/span&gt;   If I click the span right after the input element has focus, the input element will lose its focus. How does the function know that it was mySpan that was clicked?  PS: If the onclick event of the span would occur before the onblur event of the input element my problem would be solved, because I could set some status value indicating a specific element had been clicked.  PPS: The background of this problem is that I want to trigger an AJAX autocompleter control externally (from a clickable element) to show its suggestions, without the suggestions disappearing immediately because of the blur event on the input element. So I want to check in the blur function if one specific element has been clicked, and if so, ignore the blur event. Lower bound for $\phi(n)$: Is $n/5  1$? Is it true that :  $\frac {n}{5} &lt; \phi (n) &lt; n$ for all $n &gt; 1$ where $\phi (n)$ is  .  Since $\phi(n)$ has maximum value when $n$ is a prime it follows that maximum value of $\phi(n)$ in term of $n$ is $n-1$ , therefore $\phi(n)&lt; n$ for all $n$. What is the best lower bound for $\phi(n)$? What is mirror lockup and what is its primary function? My SLR is equipped with the mirror lockup function. What does it do, and when would I employ such a feature? How can I attach a file to a question? I would like to port Ubuntu Touch to a device that is currently not supported. What are the requirements for doing this and where do I begin? I have just updated (through Software Update) to Ubuntu 16.04 LTS from 15.10. There are random instances of screen flickering throughout usage. After noticing this issue, I installed Compiz Config Settings Manager, selecting Force fulls screen redraws (buffer swap) on repaint, which did nothing.   Is there any way this could be related to compiz settings or is this another issue?    EDIT: not fixed after clean install, upgrade to kernel 4.5.2, however screen flickering is reduced, it is every now and then (10 minutes apart). I will see if upgrading to kernel 4.6.0-994-generic will fix this.     EDIT: fixed as of kernel 4.6.2 generic,  currently running 4.7.2 without any issues. Definition of $\mathbb{R}^\infty$ UK Visa Refusal: Provenance of funds/parking I have applied recently for a general visitor visa to the UK to spend 10 days during Christmas with my British unmarried partner and it has been turned down.   The Entry Clearance Officer stated on his refusal notice the following:     I note that you have never previously travelled to the UK or any country with similar immigration requirements and you therefore do not have a history of travel and compliance to consider.      You have submitted an employment letter to confirm your occupation in KSA but I am required to also assess your personal, domestic and economic circumstances in determining whether you are genuine visitor travelling to the UK for the period and purpose you state.      You claim an income of SAR ........ per month which equal ........ GBP and from the evidence submitted this appears to be your sole source of income.   I note that this monthly income is reflected on your bank statements. However, the bank statement you have presented show a recent large deposit that are in excess of your stated income and inconsistent with the account history. You haven't provided evidence toward the origin of this recent deposit and in the absence of reasonable evidence I am not satisfied that these bank statements accurately reflect your personal circumstances.      In view of the above, I am not satisfied you have demonstrated that your circumstances are as you claim and therefore I am not satisfied that you're travelling to the UK for the period and purpose you stated. 41(i)and(ii).   Its true that I have deposited that sum of money right before I apply and it was a commission I received from work earlier this month.   So, could you please give me your advice on the above as I am planning to apply again soon. and how to avoid my application being refused again? Proving that the Intersection of any Collection of Compact Sets is Compact I was trying to do this problem this way:  Let $\mathcal{B}=\{B_i\}$ be a collection of compact sets. By Heine-Borel, each of the $B_i$'s are closed and bounded. We already know that the intersection of a collection of closed sets is once again closed, so $\cap \mathcal{B}$ is closed.  If I can prove $\cap\mathcal{B}$ is bounded as well, I can use Heine-Borel to say it is compact. So I was not sure how to show it was bounded, but a friend showed me his way of doing it. He did it the following way:  Let $b,a\in\mathbb{R}$ be an upper and lower bound for $B_1$, respectively. Since $\cap\mathcal{B}\subset B_1, a$ is a lower bound for $\cap \mathcal{B}$ and $b$ is an upper bound for $\cap\mathcal{B}$. Therefore, $\cap\mathcal{B}$ is bounded and then by Heine-Borel, it is compact.  My question is on the paragraph right above. I don't understand why what he did means that $a$ and $b$ are lower and upper bounds for $\cap\mathcal{B}.$ Can anyone explain why this is? Thanks. As a lab assignment I have to replicate experiment with spring oscillations, and eventually calculate spring constant. The measurements I took are: spring displacement from equilibrium $x$, mass of weights $m$, and period $T$. And presumed uncertainties: $\Delta_\alpha(x)=1[mm]$, $\Delta_e(x)=2[mm]$,  $\Delta_\alpha(t)=0.1[s]$,  $\Delta_e(t)=1[s]$. Where $\alpha$ stands for uncertainty of used equipment, and $e$ is readings uncertainty made by me.  I calculated the standard uncertainty using this formula: $$\sqrt\frac{\Delta_\alpha(x)^2+\Delta_e(x)^2}{3}$$ Which produced $u(x)=1.29[mm]$. Then using the following substitutions:  $$F=kx$$ $$mg=kx$$ $$x=\frac{g}{k}m$$ I assert that:  $$tan(a)=\frac{g}{k}$$  Because in this part of experiment I plotted $x=f(m)$, using excel's linest, I get the slope $s$  $$tan(a)=\frac{g}{k}=s$$  Now, my question is, since I had uncertainty for $x$ introduced, and it is used to compute $tan(a)$, then the latter is affected and should have uncertainty of itself? How in this case should I go about propagating uncertainty to $tan(a)$? Lost space after removing bootcamp So I removed bootcamp and now there is a partition called "free space" that I can't remove.  Here is my output from diskutil list:    Here is the screenshot from disk utility showing the partitions:    And here is the error I get when I try to remove "free space":    Heres the output when I run   sudo diskutil apfs resizeContainer disk0s2 0 I'd like a couple of Atmega328s to run the same program, but they're going to interface with each other and need unique IDs. Is there any such ID or serial number hard-coded on the chip? Evaluating even binomial coefficients Can someone give me a hint how to evaluate $$ \binom{n}{0}+\binom{n}{2}+\cdots+\binom{n}{o(n)},$$ where $o(n)$ is $n$ if $n$ is even and $n-1$ otherwise ? Can you lose an ability if your reputation falls below a threshold? I have an app to deploy in production that uses 'honor system' security. That is, all users connect to the DB using a SQL user/passwd credential and the app manages permissions itself. The latter part doesn't bother me as much as the fact that the connection object contains embedded credentials and can be copied around freely. I'm try to find some way to limit connections to a more limited set of clients. I can create firewall rules to limit by IP, of course. Is there any way to 'prequalify' SQL logins either by Machine account or domain membership? I had two drives partitioned the same and running two RAID partitions on each.  One died and I replaced it under warranty for the same model.  While trying to partition it, the first partition can only start on sector 2048, instead of 63 that was before. Drive have different geometry as previous and remaining ones. (Fewer heads/more cylinders)  Old drive:  $ sudo fdisk -c -u -l /dev/sdb   Disk /dev/sdb: 2000.4 GB, 2000398934016 bytes 255 heads, 63 sectors/track, 243201 cylinders, total 3907029168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x000aa189     Device Boot      Start         End      Blocks   Id  System /dev/sdb1   *          63   174080339    87040138+  83  Linux /dev/sdb2       174080340   182482334     4200997+  82  Linux swap / Solaris /dev/sdb3       182482335  3907024064  1862270865   fd  Linux raid autodetect   Remanufactured drive received from warranty:  $ sudo fdisk -c -u -l /dev/sda  Disk /dev/sda: 2000.4 GB, 2000398934016 bytes 81 heads, 63 sectors/track, 765633 cylinders, total 3907029168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x000d0b5d     Device Boot      Start         End      Blocks   Id  System /dev/sda1            2048  ...   Why is that? How to extract the decimal part from a floating point number in C? How can we extract the decimal part of a floating point number and store the decimal part and the integer part into two separate integer variables?