Boundary of union of two sets equals the union of their boundaries
The boundary of union is the union of boundaries when the sets have disjoint closures
Find the limit of $\frac{\log(n+1)}{\log(n)}$ where $n\rightarrow\infty$. Here $n$ is a natural number so I guess we can't use L'Hopital Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.   The  $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative system $R\setminus I$) gives a morphism $$ f_1:\operatorname{Spec}(R_I)\to \operatorname{Spec}(R)=:X $$ The  $\widehat{R}_I$ of $R$ at $I$ gives a morphism $$ f_2:\operatorname{Spec}(\widehat{R}_I)\to X $$ For the special case $I=(a)$, the localization $R_a$ (i.e. the localization of $R$ at the multiplicative system $\{1,a,a^2,\ldots\}$) gives a morphism $$ f_3:\operatorname{Spec}(R_a)\to X $$   My question is:     What is the geometric meaning of all three constructions and how are they related?   This is what I ''know'' already or describes at least the style of answer I would appreciate. As remarked in the comments, $I$ has to be a prime ideal.   $R\to R_I$ is injective iff $R\setminus I$ contains no zero divisors which is the case if $I$ is a prime ideal. The scheme $\operatorname{Spec}(R_I)$ is the intersection of all the neightbourhoods of $I$ in $X$. This is a little contra-intuitive for me since the last statement sounds like ''$f_1$ is injective'' but the $\operatorname{Spec}$-operator should turn 'injective'' and ''surjective'' around somehow (I know this is literally not true but I only want to get a feeling like ''is contained'', ''is bigger'', ''is smaller'', etc.). $\operatorname{Spec}(R_a)$ is somehow the opposite of $\operatorname{Spec}(R_{(a)})$ (= the intersection of all the neightbourhoods of $(a)$ in $X$) because $\operatorname{Spec}(R_a)$ seems to be something like the union of all the open sets of $X$ not containing the point $(a)$. $R\to\widehat{R}_I$ is injective iff $\cap I^n=(0)$ and this holds very often (e.g. for $R$ noetherian and either an integral domain or a local ring). Hence (this is probably false intuition as remarked before), $f_2$ should be something like a ''projection'' (from something ''big'' into something ''small''). But what is geometrically the difference between localization and completion. I don't have a geometric idea of completion at all.   As remarked by Qiaochu Yuan in the comments below, one should not think of $\operatorname{Spec}$ as injective-surjective switching. How can I find out which app is putting ads in my notification bar? Possible Duplicate:          Recently advertisements have started appearing as notifications in my phone's notification bar. The icon on the ad are either a green plus sign or a white star with a red circle inside it. There is no indication in the notification as to which app it came from.  I haven't installed any new applications, so I assume that some application updated to include this intrusive new ad system. Ads only show up at irregular intervals once every few hours, so removing all my apps one by one until the ads stop is not really a viable option, since I would have to wait several hours in between each uninstall to see if another ad appears. (I'm sure this is by design.)  So is there any way to figure out the source application of these notification ads?  My phone is a rooted/unlocked/whatever HTC Droid Incredible running MIUI. Best books for mathematical background? Does a Rogue using Use Magic Device for a scroll or wand also need to emulate a class feature? I'm confused about Rogues and Use Magic Device. I've searched the core book and the Internet for the past 4 hours and can't find the answer to my simple question.   Since a Rogue doesn't have any spells, a spell list or caster level, do you have to emulate a class feature in addition to all other necessary checks to use items like scrolls and wands? The question is, what elements on a DM screen are essential? I'm new at playing and DMing, altough a lot of experience watching games, just got a group to play with now. Started with the starter set in 5E and the Lost Mine of Phandelver adventure, however I've noticed the game can come to a stop when checking the rules or how to do something, while getting the books and stuff like that. So I thought to make a DM screen with some cheat sheets, of the most usual things new players, and new DM's have doubts on, and how to make the game flow well without long rule-checking breaks. What are the essential things to have on that screen? So far this is what I found  but not sure if it has all I need or so much stuff that'll also make the game stop a lot while cheking it out. Sounds of supersonic objects Would you, as a stationary observer, hear a sound in reverse when the source of the sound travels with twice the speed of sound? Of course, he wouldn't hear anything at all before the airplane passed by and the sound reaches him, but what about after? Does it reverse due to the Doppler effect? Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer?  The square roots need to be irrational. Outcast boy meets girl underwater Displaying world country shapefiles centered on Pacific Ocean using Robinson or Miller Cylindrical projection in QGIS? I wish to display a map in QGIS (world country shapefiles) showing all countries but centered on the Pacific area.   I am not familiar with Proj4, so is there any way this can be done in QGIS? My internet connection is very slow. I suspect someone in the network is using a BitTorrent client which takes huge amount of bandwidth.  Is there a way to detect who is using my bandwidth (network sniffing tools etc)?  I should have pointed out that I have no access to the router, what I am looking for is a way of detecting someone's MAC/IP if she uses Bittorrent. What are some examples of induction where the base case is difficult but the inductive step is trivial? According to :     ...proofs by mathematical induction   have two parts: the "base case" that   shows that the theorem is true for a   particular initial value such as n = 0   or n = 1 and then an inductive step   that shows that if the theorem is true   for a certain value of n, it is also   true for the value n + 1. The base   case is often trivial and is   identified as such, although there are   cases where the base case is difficult   but the inductive step is trivial.   What are some examples of proofs by induction where the base case is difficult but the inductive step is trivial? I clean-installed Ubuntu 11.10 today, and then installed VirtualBox.  This required me to add myself to the vboxusers group, and since 11.10 seems to no longer have a graphical app to add users to a group, I ran the following command:  sudo usermod -G vboxusers stephane   This is a problem.  I now see what I should have run instead is:  sudo usermod -aG vboxusers stephane   The end result is I'm no longer in the groups I should be in.  Including whatever group is required to run "sudo".  When I run any command as sudo now, I get the following:  $ sudo ls [sudo] password for stephane:  stephane is not in the sudoers file.  This incident will be reported.   Is there a way to fix this, or do I need to re-install from scratch again? We all know you can't do the following because of ConcurrentModificationException:  for (Object i : l) {     if (condition(i)) {         l.remove(i);     } }   But this apparently works sometimes, but not always. Here's some specific code:  public static void main(String[] args) {     Collection&lt;Integer&gt; l = new ArrayList&lt;&gt;();      for (int i = 0; i &lt; 10; ++i) {         l.add(4);         l.add(5);         l.add(6);     }      for (int i : l) {         if (i == 5) {             l.remove(i);         }     }      System.out.println(l); }   This, of course, results in:  Exception in thread "main" java.util.ConcurrentModificationException   Even though multiple threads aren't doing it. Anyway.  What's the best solution to this problem? How can I remove an item from the collection in a loop without throwing this exception?  I'm also using an arbitrary Collection here, not necessarily an ArrayList, so you can't rely on get. Quick edit not recorded as such blocking vote change A little background:  I do not know what the specifics are, but I've noticed that a quick edit after a post will often times not result in an edit being posted. It's as if there was no edit. Edit: According to Bart this amount of time is five minutes.  The problem:  A user made a really bad post (said that ~ in C# wrt integers was used for deconstruction). I down-voted said post. Post quickly disappeared (I guess poster realized they screwed up and took back the post). Later, I come back and see my down vote but the answer is now correct (that ~ is the bitwise complement operator). I try to get rid of my down vote because it is no longer appropriate. However, it has been about half an hour and there is no edit posted (presumably the poster made an edit within the non-posting edit window) so I can't take my vote back.  Suggested solution:  If post gets edited (even within the non-post edit window) mark all the votes as changeable. I would guess that this would either involve an update on a table of votes or adding a column for last edit time which is not bypassed by the grace period. I realize that this may be an entirely non-trivial change, but unless the non-posting of edits is somehow changed to make sure the edit was trivial it seems this is necessary. Unit testing code which has logic around the CreatedDate An example of the problem would be trying to test a trigger on a case which will fire on update, and look for time differences between when a case was created and now. Ideally in the unit tests I want to create a case with a CreatedDate in the past. The problem here is that CreatedDate is an audit field and not writeable[1]. Is there a standard work around to this problem?  [1] I know you can get audit fields turned on to be writeable temporarily when doing data migration. This is not a viable solution for every time you want to get your code coverage up for deployment. I am a bit financially strapped and received a Gateway GT5056 tower. The tower has an AMD dual core 64-bit processor in it. Will Ubuntu 12.04 run on this machine? I will not have any other application other than LibreOffice installed. Meaning of Sun Lamp Size? In Cycles, a sun lamp is treated as infinitely far away - only the angle matters, not the position. Other lamps have their sizes defined in blender units: a point lamp with size 1 emits light from a area with size of 1. But if a sun lamp at infinity had a size of 1 blender unit, the angular arc of that sun would be nonexistent.  Clearly, when you change the size of a sun lamp, shadows get fuzzier or sharper. But what does that actually mean? What are you changing? $K \subset \mathbb{R}^n$ is compact iff it is closed and bounded I want to prove $\Leftarrow$.  I just need one more step, now. If I find a compact set that contains K, then it will finish the proof. Rudin take a k-cell to do that, but before it he proves that every k-cell is compact, and I think I can take a more simple compact to do this step. I has thinked in the closed ball, but don't know how to prove that it is compact. Can anyone help me to find this set, if it really exists? Thanks. Using CLI, how do I change the user's profile picture? A similar task for changing the desktop background, can be achieved using the following script:   FILE="/backgrounds/background.png" gsettings set org.gnome.desktop.background picture-uri "$FILE"   How would I achieve the same result, but with the user's avatar/profile picture?