Minimal integer to make a rational into an integer
Compute the period of a decimal number a priori
Which one sounds better? "of mine" or "my" My friend (or "a friend of mine"?) told me that anytime I need to express such expression, it's always better to say "of mine", because it sounds more naturally.  Like "two brothers of mine" rather than "my two brothers".  Is that true? Convergence of exponential Brownian martingale to zero almost surely Define the exponential Brownian martingale as $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ which is a martingale with respect to the natural filtration of $W$ which stands for a standard Brownian motion.  I now want to prove that $N_t \rightarrow 0$ a.s. for $t \rightarrow \infty$. As $N_t$ is a martingale it is by definition in $L^1$ hence Doob's martingale convergence theorem tells us that there exists a $N_\infty \in L^1$ such that $N_t \rightarrow N_\infty$ a.s., but how could I prove that $N_\infty \equiv 0$? Any help is greatly appreciated! Some areas of the mesh are dark (Shadow artefacts) I've been modeling a hermit crab and I don't know why but some areas of the mesh are really dark on some spots, like the legs and the claws.   I hope you can help me fix it. I am a new user for Ubuntu. I downloaded 13.10 yesterday and made a bootable USB with universal USB installer on my Lenovo IdeaPad Z580 laptop which is now using UEFI; everything appeared OK.  When I booted from the LiveUSB I got the choices of trying or installing Ubuntu but both of them keep giving me these error:  [ 1.929082] kernel panic-not syncing vfs unable to mount root fs on unknown-block (0.0)   Please tell me what is going wrong? Changing the size of the mouse pointer in Ubuntu MATE 18.04 does not work I have Ubuntu 18.04 with the gnome-desktop  and parallel the Ubuntu MATE desktop installed. While I can easily change the size of the mouse pointer (DMZ-White) in the Gnome3 desktop , this does not work on the MATE desktop. I use for this the dconf-editor.  Maybe someone knows a solution?  Thanks a lot. Can the Artificer Artillerist's tiny-sized Eldritch Cannon that is not held in hand still ride or be carried elsewhere on the body? Change Proxy Settings In Ubuntu I want to use the Ubuntu Software Center to install software, but I cannot download them.  How can I give my proxy settings in Ubuntu terminal (just like we use cmd in Windows to choose IE proxy settings)? Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$. Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.   I do not understand how to go about completing this problem or even where to start. How do I merge 2 VMware disks in to a single one? I have installed an Ubuntu server on a VMWare workstation with a space of 20GB storage as seen below:    After the installation I've attached another 40GB worth storage to the existing Ubuntu server installation (seen below):    Now I want to add this 40GB to my / (root) directory it mean's  I want that my / directory became to 60G , like it will be resize from 20G to 60G (20G + 40G), i don't know what should i do, can any body help me ?  , my df -h result shows;  root@Userver:~# df -h Filesystem      Size  Used Avail Use% Mounted on /dev/sda1        18G  900M   16G   6% / udev            990M  4.0K  990M   1% /dev tmpfs           400M  308K  399M   1% /run none            5.0M     0  5.0M   0% /run/lock none            999M     0  999M   0% /run/shm none            100M     0  100M   0% /run/user root@Userver:~#   it must be 60G !  i have no graphic mode, then i can't use Gparted tools, i'm just have ssh access ! Absolutely continuity and Lipschitz-like continuity I'm trying to figure out if the following is true or false:  Let $\mu$ and $\nu$ be finite measures on Borel subsets of $[0,1]$,and $\nu \ll \mu$ ($\nu$ is absolutely cts with respect to $\mu$). Does there exist always a $C \in \mathbb{R}$ such that $\forall A \subseteq [0,1]$ that is Borel, we have $\nu(A)$ $\lt$ $C \mu (A)$.  The Radon-Nykodym theorem seems to almost suggest that is true, if I knew it was bounded, then I could say $\nu (A) \le \int_{A} g \ d\mu \le \int_{A} C \ d\mu \le C\mu (A)$, where $C$ is the bound. However, I don't know if it is bounded and in fact, I could take an unbounded function that is still integrable on $[0,1]$, such as $\frac{1}{\sqrt x}$.  I can't even be sure if it is true or false.   Thank you! How to understand linux versioning I need to understand the Linux versioning system and distribution concepts. what are public, stable or final terms for versions? Should I use 'or' or 'nor' after a negative statement? How to correctly post a Q&A style question (i.e.self answer) without it failing minimal understanding As I understand it Q&amp;A style questions are  but a self answered question should be judged independently of the answer. Often after I've solved a problem I would like to share the question and answer on stack overflow (and have on a couple of occasions) but how to show &quot;minimal understanding of the problem&quot; is always challenging. For example I have been looking into the following question  How can I plot a straight line in the JMonkey Engine library I'm trying to plot straight lines between vertices that I specify using the JMonkey Engine 3D graphics library. So for example if I was to try to plot between: (2,0,0) (-1,0,1) (0,1,1) (1,1,1) (1,4,0) Then I would get:   This is something I do (and forget how to do) relatively often and the answer is harder to find than you'd imagine ( provides everything needed to figure it out but its spread out over several posts and isn't a complete solution on its own). Therefore I want to provide a Q&amp;A for it but at present this question feels like it would fail the minimal understanding test  Questions must demonstrate a minimal understanding of the problem being solved. Tell us what you've tried to do, why it didn't work, and how it should work  Should I make up some imaginary problem? (Probably the final source code messed up a bit) Or mention in the question that I’m self answering to assuage some of the normal &quot;This is not a do your work for you site&quot; comments. Maximum of $|\sin(z)|$ as $\{z: |z| \leq 1 \} $  So according to the Maximum Principle, the maximum is when $|z|=1$. I tried using the fact that $\sin z= \dfrac {e^{iz}-e^{-iz}}{2i}$, but didn't know how to continue from there. Setting $z=e^{it}$ doesn't help much.  I saw this question was answered before, however the answer uses hyperbolic functions ($\sinh, \cosh$) which I haven't actually studied.  Thanks in advance for any assistance! I have a simple question about calculating distances in PostGIS.  I would like to get the distance between two geometries. I am using this sid : 4269 in meters what i am doing now is this : ST_Distance((a.geom,b.geom)) FROM ... but I am getting result in degrees. I think that I should work with geography but how can I cast a geom to a geography? I tried with (a.geom::geography) but i am getting an error.  What can I do to get my result in meters?   Thank You When does the Death Saving Throw happen with respect to sources of automatic ongoing damage? If a character is knocked unconscious with 0HP with a source of automatically dealt damage, for example, into a fire, and takes damage from it each turn, does the automatic Death Saving Throw failure from taking damage at 0hp happen before or after the player's regular Death Saving Throw roll?  If the unconscious PC takes the auto-fail first then they effectively only get 1 Death Saving Throw to survive, because they'd autofail and roll a failure the first turn, then autofail again and die the next turn.   If they get to roll first, then they only effectively get two death saving throws before they die. How can I have `date` output the time from a different timezone? I have a server running with the timezone set to UTC. It seemed like that was generally a good practice (please correct me if I'm wrong).  Anyhow, one of the servers I connect to, in order to scp files, is running on EDT and stores files that I need to copy in the format /path/to/filename/data20120913  I looked at trying to rsync files using something like find's -mtime -1 flag for files modified in the last day, but I didn't have any luck.  I don't mind just using scp to copy the current day's file, but as of right now there is a 4-hour window where running date +%Y%m%d will give a different day on each server and that bugs me a little.  Looking through man date I see that I can have the time output as UTC, but I don't see a way to have it output as another timezone like EDT  I suppose I could also use something like the GNU date extension date -d 20100909 +%s to get the date in seconds from the epoch, apply a manual 4 * 60 * 60 second calculation, and see about rendering that as a date - but then when daylight time kicks in it will still be an hour off.  Is there a simpler way to output the date in a YYYYMMDD format for EDT on a server that is set to UTC ? This question is inspired by my own answer to a  which I tried to answer and got stuck at one point.     The question was:     HI DARLING.      USE MY ATM CARD, TAKE ANY AMOUNT OUT, GO SHOPPING AND TAKE YOUR FRIENDS FOR LUNCH.      PIN CODE: $\displaystyle \int_{0}^{1} \frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 - 3x + 2}} \, dx $      I LOVE YOU HONEY.      Anyone knows? Are we gonna get an integer number?     My attempt:  Does this help?     $$\frac{3x^3-x^2+2x-4}{x-1}=3x^2+2x+4$$   (long division)   \begin{align*} I&amp;=\int\frac{3x^3-x^2+2x-4}{[(x-1)(x-2)]^{1/2}} dx = \\ &amp;=\int\frac{(3x^2+2x+4)(x-1)^{1/2}}{(x-2)^{1/2}} dx = \\ &amp;=\int 3(u^4-4u^2-4)(u^2+1)^{1/2}du \times 2 \end{align*}   after the substitution   \begin{gather*} (x-2)^{1/2}=u\\ du=\frac1{2(x-2)^{1/2}}dx\\ u^2=x-2\\ (x-1)^{1/2}=(u^2+1)^{1/2} \end{gather*}   Update:  may help us proceed.    I tried to proceed:     $$6\int (u^4-4u^2-4)(u^2+1)^{1/2} du = 6\int ((t-3)^2-8)t \frac{dt}{2u}$$   after $u^2+1=t$ and $dt=2udu$   \begin{align*} u^4-4u^2-4 &amp;= (u^2+1)^2-(6u^2+5) \\ &amp;= (u^2+1)^2-6(u^2+1)+1 \\ &amp;= ((u^2+1)-3)^2-8 \end{align*}   I wonder whether this question can be solved from here?    Update:   This has been getting a lot of views, and I think most people came for the sort of problem mentioned in the title (where I got stuck) rather than the original problem itself.  Keepin this in mind, I'm reopening the question and here's the kind of answers I expect — Solutions to the original problem are good, but I'd prefer solutions that continue from the part where I got stuck — the polynomial in $u$ — that's the sort of problem mentioned in the title. I was looking for a possible duplicated question about  in , then looked using the Big G! since the integrated search didn't give me any useful result (laugs) then I found the , but in Stackoverflow! Then, I proceed to flag as off-topic, but since "It's too old to migrate" I don't have the option to do so...  So, my question is, should I leave it like that? Or the question is better in UL (also on topic, just read the question)? Any guidance to this young pada-wan is greatly appreciated. When editing a post and using the image uploader, any unused link references are completely removed.  For example, this post has three image references, only one of which is used ([3]) in the body of the post:        After uploading a fourth image, references [1] and [2] are completely removed (and the remaining references are renumbered to become [1] and [2]):        I sometimes write link references before using them in the body of the post and sometimes remove images temporarily before reinserting them elsewhere in the post so this is inconvenient.  (Note: this also applies for the iOS app, having the same behavior)