Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?
Can you partition an infinite set, into an infinite number of infinite sets?
Longest line using awk I am wondering if there is a modifier or tool that will essentially undo a subsurf or subdivide command. Experimental data for asymmetric Newton cradle Using a "successive impact model" (as if each ball were seperated from the other ones), I produced the following animations:    You can see any combination of balls with masses of 1 or 2 (left) or 1 and 4 (right).  Unfortunately, I do not have any Newton cradle to make some experiments, and I'd like to compare my results with observations. I contacted the author of , which is the most instructive one I found about asymmetric Newton cradle. In particular, he writes that the "OoO" configuration behaves as the "ooo" one, which is not what I simulate. But the author is not sure whether who is wrong because he did the experiments a long time ago.  I am especially interested in asymmetric Newton cradle with simultaneous impacts: left and right balls collinding at the same time.   Please let me know if you have any idea on where to find such experimental data.  Edit Just of few general remarks to avoid extending the comments.    Energy and momentum are conserved. However, they are sufficient to ensure uniqueness of the post-impact velocities only when there are 2 balls. The successive impact model precisely consists in propagating the impact ball after ball, and therefore leads to a unique solution (which conserves momentum and energy globally). From the waves point of view, changing the mass of a ball can be done by increasing the diameter, or the density (or a combination of both). This simple model cannot account for such subtleties of course, but maybe it does not matter when the balls are "small enough". Even if I think it's quite basic, following WetSavannaAnimalakaRodVance's comment, I am ready to hand out my Mathematica code to anybody who wants it. I'll check for good ways to share it. understanding "mount" as a concept in the OS I want to understand what mounting is. It is used in different contexts and situations and I can't find resources which:   Describe the mount concept Explain the actions taken by the computer/OS/utility when a mount is performed How and in which situations mount is used Which features in the Linux mount command are of frequent use and some examples   ( I hear mount applied to diverse entities directories, flash drives, network card, etc ) $K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded. I need to prove that $K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function  in $K$ is bounded.  One direction is obvious because of Weierstrass theorem. How can i prove the other direction? I tried to assume the opposite but it didn't work for me.  Thanks a lot. I have very little background in physics, so I apologize if this question is painfully naive.  Consider the following thought experiment: an observer is in a closed room whose walls, floor, and ceiling are made entirely of mirrors, with a single light source in the middle of the room. When the light is on, the observer can see many copies of his reflection all over the place.  Suddenly, the light source turns itself off. Intuitively, I would expect the observer to "instantly" see darkness. However, I can't figure out why that is the case under the "particle" interpretation of light. There are obviously lots of photons already in the room from before. Furthermore, we know that they don't get "consumed" when they hit a wall, because otherwise the observer wouldn't see so many reflections of himself. Basically, when the light goes off, what happens to the photons already in the room?  I suspect the answer goes something like this: the photons in the room lose a little bit of energy every time they bounce off a mirror, but it's so minuscule that we can still see more reflections than our eye can resolve anyway. When the light goes off, however, it takes them a very small fraction of a second to bounce around the room enough times to diffuse completely, which our eye cannot detect.  Is that about right? If we had a theoretical "perfect reflector", would the light remain trapped in the room forever? If we had instruments that could measure such things very finely, would it take (slightly) longer for the light to go out in a room made of mirrors as opposed to a room made of, say, black cloth? Making overview with linked detailed maps in QGIS? i want to create 4-5 detailed maps, and then an overview map that has rectangles that markes the locations of the detailed maps. Best case would be if the rectangles were links to the detailed map.   I have managed to make the overview and the rectangles, and think I know how to make the detailed maps one by one. The rectangles are drawn manually on the overview map. All layers that I plan on using are in the same QGIS project.  The final product that I hope to create would be the overview map on the first page, followed by the detailed maps. The overview would have links to the detailed maps, click on the rectangles to open the right detailed map.  Is there a way to link the detailed maps to the overview in QGIS 2.8 print composer?  I'm still not sure if I'm doing something wrong, when I try to create the overview. I want it to show the extents of all of my detailed maps.   Is this possible? I am interested in theoretical and practical considerations. Cannot find Background Images feature in Blender 2.8 Cannot find Background Images feature to add in a photo in Blender 2.8  Where is it? Were there other marital relationships besides Anakin and Padme in the Jedi order? There is a vague idea that Obi-wan and Queen of Mandalore had a thing, but this isn't properly addressed in the Clone Wars series AFAIK.  But are there other instances of Jedi who go against the order for love/relationships? Can anyone explain IEnumerable and IEnumerator to me? Can anyone explain IEnumerable and IEnumerator to me? For example, when to use it over foreach? what's the difference between IEnumerable and IEnumerator? Why do we need to use it? How do you use Batch for send more than 10 emails? Is there a way to block a list of specific apps on macOS?  I have looked up solutions such as Cisdem, but there are some holes in the way it works that I don't like.   Unless you also block Activity Monitor, Terminal, and System Preferences, you can simply quit the Cisdem process or create a new user to get around it. You have to block Activity Monitor, Terminal, and System Preferences to make it work well.   I want to block an app from running or even being installed on my Mac. The app can be installed via the App Store and the web. Not sure how to go about this. Enabling Parental Controls won't work as Administrator privileges are needed.   The specific app I want to block is Apple Configurator.  Any ideas? What are your thoughts on epigraphs in theses? I am wondering if there is a general consensus as to the suitability of epigraphs in a thesis, either a single one at the start of the document, or an appropriate quote to begin every chapter.  For example, at the start of a technical chapter describing some code, I could write     Beware of bugs in the above code; I have only proved it correct, not tried it. - Donald E. Knuth   I am lucky to have complete freedom over how I choose to typeset my thesis (Master's thesis in astronomy) and I like the idea of a thematic quote to begin a chapter. However, I am wary of seeming unprofessional or tacky so I am curious to get some more thoughts on the matter. How is Angular Momentum Conserved when Mass is Released? I am not a physicist (math/comp-sci) but I understand that Angular Momentum is supposed to be conserved.  I find this confusing because there seems to be many simple, common cases where a restrained, revolving body, when released seems to lose it's angular momentum.  For example, let's say that a hammer thrower is revolving a 10kg hammer around herself at a 1 meter distance from their combined center of mass every second.  Taking the hammer as a point-mass, then before release, the hammer and the thrower each have an angular momentum of      $l\omega = (mr^2)*(2\Pi) = (10*1*1)*(2*3.14..) =~ 62.8 kg\text{-}m^2/s$   Now, after the hammer is released, the thrower still has her same angular momentum (also 62.8), but the hammer no longer seems to have any.  Yes, I know that the hammer is still rotating (spinning) as it did when it was revolving around the thrower, so that angular momentum seems to be conserved, but the angular momentum from revolution (orbiting) seems to be gone.  AFAIK, it's not transferred to it's spin angular momentum (observation seems to bear this out).  Nor is it transferred to the thrower.  Besides having no mechanism for this (she let go of it), if it did, we would have noticed hammer throwers being knocked over by having their angular momentum suddenly doubled.  So where does it go?  Or is it not actually conserved in this case? Why are there greyed / dimmed / faded questions now? I just refreshed the [iphone]-tag-site, and I'm seeing all kinds of questions with what appears to be a grey overlay (see image). What is this for/why am I seeing this? I have never seen anything like this before.. 16.04 LTS upgrade alert- How to handle it? I am running Ubuntu 14.04 LTS. Today when I logged into my computer I got an update alert saying that I could upgrade to Ubuntu 16.04 LTS. The other options were not now or do not upgrade. If I chose to upgrade will it delete my /home directories and the installed software under /opt and /usr/local ? Why does "ls -all" show time for some files but only year for others? If you issue ls -all command some files or directories have the year and some have the time? Why do some show the year while others show the time? Is the time representative of the time the file was created? Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. To reopen a closed question,  must agree that the question is suitable for the site and cast votes to reopen the question. But:  How does one actually vote to reopen? Is there a reputation level that you must have before you can see this function? Or do I just leave a comment along the lines of, &quot;I vote to reopen&quot;? What happens if someone with less than 3000 reputation leaves a comment that they wish to vote to reopen? How can one draw attention to their closed questions? Is it done via a posting new question? Or do you edit the closed question?   Related:  For more information, see &quot;&quot; in the .