Why is$ (1+\frac{1}{n})^n=e$ when n goes to infinity?
How come such different methods result in the same number, $e$?
Why does Google not consider dot in usernames of Gmail addresses? E.g.: If your Gmail ID is abc.xyz@gmail.com, it considers this the same as abcxyz@gmail.com.  Why is this so? Are vacuum fluctuations really happening all the time? Is this a correct demonstration for why elements above untriseptium cannot exist? With the confirmation that elements 113, 115, 117, and 118 are indeed fundamental elements that are now to be named on the periodic table, the next question is: what is the highest atomic number possible for an element? Feynman had a go at this years ago, and he derived (according to my limited understanding) that above element-137 (informally denoted 'Feynmanium') the electrons in the nearest orbit about the nucleus would be traveling at a velocity greater than the speed of light. (Note I am considering the Bohr model of the atom in this question.) I wished to demonstrate to my friends why this is and I came up with the following explanation.  According to Bohr's quantum condition, the angular momentum of an electron about the nucleus is given by $$ L=m_evr_n=n\hslash \implies r_n=\frac{n\hslash}{m_ev},$$ where $m_e$ is the mass of the electron, $v$ its velocity, $n$ an integer, and $r_n$ the radius of the $n$th possible orbit. Because we are concerned with the nearest orbit to the nucleus, $n=1$; and thus $$r=\frac{\hslash}{m_ev}.$$   Now, according to Coulomb's law, if the electron orbits about the nucleus the centripetal motion can be described by $$\frac{Ze^2}{4\pi\epsilon_or}=m_ev^2, $$ where $Z$ denotes the number of protons in the nucleus (the atomic number), and $e$ the elementary charge. Solving for $Z$ and substituting in $r$ from above yields $$Z=\frac{4\pi\epsilon_o \hslash v}{e^2}. $$ But what is $v$? Well, the maximum velocity an electron could ever have is the speed of light, and we wish to find the atomic number associated with an orbiting electron traveling at this speed, so we set $v=c$ and obtain our final result of $$Z=\frac{4\pi\epsilon_o\hslash c}{e^2}\approx 137.521,$$ which implies that for $Z&gt;137$, the electrons at a position of $n=1$ in the Bohr model would have a velocity $&gt;c$; and thus the highest atomic number achievable on the periodic table is 137.  Again, I just want to make sure this is a correct method for deriving element-137 before presenting. Perhaps one could explain how relativity plays a role here. I know Feynman used the Dirac equation to get this result...so could anyone (subsequently of course) expatiate on this in a simplistic manner? Thanks! Convert /etc/network/interfaces to netplan for floating IP I am new to the netplan on Ubuntu 18.04, just discovered it exists today. I have an interface that I am trying to add to create a floating IP on google cloud. It is based on this example:   I tested the example, and it works, but now I try to do the same on ubuntu, and I don't know how to convert this:  cat &lt;&lt; EOF &gt;&gt; /etc/network/interfaces auto eth0:0 iface eth0:0 inet static     address 10.190.1.1     netmask 255.255.255.255 EOF   to netplan.  The output for ls /etc/netplan is '50-cloud-init.yaml'. and the output for cat /etc/netplan/*.yaml is:  network:     version: 2     ethernets:         ens4:             dhcp4: true             match:                 macaddress: 42:01:0a:8e:00:3e             set-name: ens4   my ifconfig:  # ifconfig  ens4 ens4: flags=4163&lt;UP,BROADCAST,RUNNING,MULTICAST&gt;  mtu 1460     inet 10.142.0.62  netmask 255.255.255.255  broadcast 0.0.0.0     inet6 fe80::4001:aff:fe8e:3e  prefixlen 64  scopeid 0x20&lt;link&gt;     ether 42:01:0a:8e:00:3e  txqueuelen 1000  (Ethernet)     RX packets 9430  bytes 1635180 (1.6 MB)     RX errors 0  dropped 0  overruns 0  frame 0     TX packets 13383  bytes 1513428 (1.5 MB)     TX errors 0  dropped 0 overruns 0  carrier 0  collisions 0   What is the equivalent of what he said to put in the file, for netplan? Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ||$\mathbf z - \mathbf x$||$_2$ over all $\mathbf z \in C$.  (a) Show that the minimum exists.  (b) Show that there is only one minimizer: that is, show that if $\mathbf y_1$ and $\mathbf y_2$ both minimize ||$\mathbf x - \mathbf z$|| then $\mathbf y_1 = \mathbf y_2$   My ideas:  For (a), to show that the minimum exists I will need to show that it is closed and bounded. We already know it's closed because C is a closed convex subset so I just need to show it's bounded. Am I wanting to minimize ||$\mathbf x - \mathbf z$||$_2$? This is just the Euclidean norm which would be $\sqrt{(\mathbf z - \mathbf x)^2}$=$\sqrt{\mathbf z^2 -\mathbf z \mathbf x -\mathbf x \mathbf z + \mathbf x^2}$. I'm not sure where to go from here...  For (b) I want to use proof by contradiction- so I will assume that $\mathbf y_1$ and  $\mathbf y_2$ both minimize the function and then I'll somehow see that they are in fact equal. How do I set up this argument? Can we omit parentheses when creating an object using the "new" operator? Analysis's output vs analysis output vs output of the analysis Replace arbitrary characters in the middle of an IP address string with sed I need to find and replace either one or two numerical characters in strings in a file.  The strings are IP addresses of the form:   10.xx.y.z    Where xx can be one or two characters.    I want to replace the xx with the single character 0, so I have  10.0.y.z    preserving the values of y and z.   The string may appear multiple times in the file.  What is the sed invocation to do this? What are some grandiloquent, or simply better, ways of expressing "an idea/thought suddenly came to me", or "an idea/thought struck me", or "I was struck by an idea/thought"? Why isn't Derivative equal to NeumannValue? Why cast an unused function parameter value to void? I created a model and added some Actions in the DopeSheet editor (more or less 5).  I exported the model to my game engine, saved the .blend file (Ctrl-S) and closed it.  Then I opened it again, and all the actions were gone, it just had the one I was using last.  The same happens with some preferences (for example, I disabled mip-maps on User Preferences -> System, and I have to re-enable it every time I open the .blend file again.  Does somebody knows how to restore the Actions and save everything? Thanks. Is there a way to condition output on a given date? I'd like to be able to print text that varies depending on whether the date matches a given date string  For example  suppose there was a command   \iftoday#1#2#3   which printed #2 if \today matched the string in #1 and #3 otherwise.  So a sample of my proposed usage would be:     \iftoday{February 3, 2016}{There's no garbage today}{Please put out the garbage}    I've tried various commands like pdftexcmd's strcmp, to no avail.    I suspect the problem may have to do with matching the string generated by \today.    While on screen it appears that \today returns February 3, 2016. I can't get it to match February 3, 2016 when I compare strings.  Obviously, I'd be perfectly happy if I could get this to work by represent today's date in any of the standard date formats.  Thanks very much in advance for any suggestions. When new users edit a post, the review is posted to a review queue and other reputable users can review the edit and choose to approve or reject it. However, I feel that a lot of people do it pretty quickly.    to . He was fixing broken links in the answer, and as soon as I saw it I knew it would have been a good edit. However, only one reviewer found so, and three others rejected it. I ended up manually doing the edit, and poor Onato didn't get any credit for it (in fact, he got an undeserved 'rejected' for it).  The point I want to bring is that, first and foremost, the person who wrote the answer probably knows best if an edit is acceptable or not. Could there be a time frame where the author of the answer (or question) can overturn a review decision?    EDIT: one more  was rejected by overeager reviewers as well. I absolutely disagree with the notion that the community knows better. The community repeatedly proves that it only wants Internet points.    EDIT: I told a commenter to fix something himself in one of my posts, but . I went back and applied the changes myself. It's not credited to him and his stats got a rejected edit now (which really should be an approved edit). The reason ("This edit is incorrect or an attempt to reply to or comment on the existing post") is blatantly wrong and shows once more that the community has much less insight into edit quality than the people who wrote the answers in the first place. It's true that the UI didn't make it easy for them, but those guys clearly didn't read the comment thread; I did, because as the author, I was notified.    EDIT: This time around, a user fixed a blatant (though small) mistake in one of my accepted answers. One hour later, another, unrelated 10k user , even though it had been approved by the community. I went back to undo his undo.  This goes on again to show that an active user is the best judge for the quality of an edit, and that since I can have the last word on edits on my own posts, I should be allowed to retroactively overturn a community edit vote and accordingly credit the person who proposed the change.    EDIT I wrote a  years ago. Some guy found it and tried to fix it but the community thought that it wasn't worth it. I think that the changes are great and once more I'll integrate them by myself into my post. Unfortunately for this user, he won't be credited for it. I still don't understand why I can't have a binding vote on edits on my posts. Random place change in row with restrictions Defer SP.RequestExecutor.executeAsync I'm working on a SharePoint app where I need to run SP.RequestExecutor.executeAsync() in order to grab some information from one of the Host Web's lists. Depending on what method is calling the oData call I want to do something different.   Unfortuntely, executor.executeAsync() isn't playing as nice as $.ajax() does in terms of using $.when().done() and executing code after an async call is made. Below is some code if you'd like to see:      load = function (){         $.when(getEntries()).done(function () {             ViewModels.Calendar.addEventSource(ko.utils.unwrapObservable(eventList));         });     }      getEntries = function () {         return executor.executeAsync({             url: appweburl                 + "/_api/SP.AppContextSite(@target)/web/lists/getbytitle('" + LIST + "')/items?@target='" + hostweburl                 + "'&amp;$select=Title,OData__x006e_ot5,qnlu,OData__x0066_x20"                 + "&amp;$filter=OData__x0066_x20 eq '" + ViewModels.Person.user.userName() + "' "                 + "and qnlu ge DateTime'" + startDate().toString("yyyy-MM-dd") + "T00:00:00' "                 + "and qnlu le DateTime'" + endDate().toString("yyyy-MM-dd") + "T00:00:00' ",             method: "GET",             headers: { "Accept": "application/json; odata=verbose" },             success: onGetEntriesSuccess,             error: onoDataCallFailure          });     },      onGetEntriesSuccess = function (data) {         var jsonObject = JSON.parse(data.body);         $.each(jsonObject.d.results, function (index, item) {             eventList.push(new Event(item.qnlu, item.OData__x006e_ot5));         });     },      onoDataCallFailure = function (data, errorCode, errorMessage) {         alert('Failed to get host site. Error:' + errorMessage);     };   Any ideas/suggestions would be appreciated. I have been playing with Ubuntu for a few weeks now, and I'd like to revert my computer back to it's original - factory - defaults.   On the computer I have a recovery partition (it's a netbook). I went through the process of recovery and everything seemed fine. However, when I restart the computer I'm presented with   grub rescue &gt;   Now, my understanding is that when I installed Ubuntu "side by side" it replaced the MBR or something like it, with GRUB.   I've read on a slew of forums, that I need to use a Windows Recovery Disk.   Here are my issues:   a) I don't have a recovery disk, I have a recovery partition - it's a netbook. b) I don't have an external cd drive.   What I do have is a USB key that has about 1gb of space on it.   Thanks in advance. Book: Handful of children kept in a school as political mythological prisoners Say, for example, you have a card that is still legal in standard but the version you have of it comes from a set that has since been rotated out. Can you still play the version from your older set? Singularities in the punctured unit disc and square integrability I'm having trouble with the following problem:  Let $f$ be holomorphic on the punctured unit disc, $D$. If $\int_D|f(z)|dA(z)&lt;\infty$, then $z=0$ is either a removable singularity or a simple pole of $f$.  A similar problem is to prove or disprove that if $\int_D|f(z)|^2dA(z)&lt;\infty$, then $f$ has a removable singularity at $0$.  I've tried using the Mean Value Property for $f$ and taking the limit as $z\to0$, but I didn't get anywhere.