How to draw the plot of a function in the complex plane?
Plotting Complex Quantity Functions
Issue with 'StringVar' in Python Program How can I show that $ab \sim \gcd (a,b) {\operatorname{lcm} (a,b)}$ for any $a,b \in R \setminus \{0\}$? Let $R$ be a UFD. Then for any two elements $a,b \in R \setminus \{0\}$ $ab \sim \gcd (a,b){\operatorname{lcm} (a,b)}$.   My attempt :  Let $d = \gcd (a,b)$.Then $d|a$ and $d|b$.Then $\exists x,y \in R$ such that $a = dx$ and $b = dy$.Now it is to be shown that $c = dxy$ is the lcm of $a,b$.Which I find difficulty to show.It is clear that $a|c$ and $b|c$. Now if $u$ is any common multiple of $a$ and $b$.Then $dx|u$ and also since $y|b$ we have $y|u$ i.e. $dxy|u^2$ $\implies c|u^2$.But I have to show $c|u$ to complete the proof which I fail to prove.Please help me to complete the proof.  Thank you in advance. Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$ Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction. I've recently started playing Minecraft, and I only have a little experience with enemies (I've been playing with the peaceful setting so I can figure out what the heck I'm doing).  When I did play some survival mode, I happened to be at the top of a pretty large cliff when night hit, and this seemed to protect me - the baddies just piled up at the bottom of the cliff.  However, it seems like this probably only worked due to pathing flukes.  The terrain behind me was pretty easy going, so I suspect if enemies spawned in the right areas, they could still come up and visit.  That said, how can I set up a quick protective shelter for the first night or two?  The days don't seem to last very long, so I don't expect I would be able to generate a lot of high-end materials, tools, etc. to work with.  Also, the zombies caught fire nicely when the sun came up, but there were still some spiders and some other things (creepers? - they were explosive) that didn't go away at daybreak.  Do I need multiple exits for my shelter or some other way to avoid running into all the enemies outside the front door that have accumulated overnight? Can I shrink the transaction log file on a mirror database? This is a followup question to  about why I couldn't shrink the log file on the principal database.  To make a long story short, I setup database mirroring but forgot to make sure the job that backed up the transaction log was running again, and the transaction log grew to almost 60GB.   Since mirroring was set up, this size increase was duplicated on the mirrored server, and eventually took up all the disk space and made the mirror database unusable.  Per  about transaction log maintenance on a mirror database, you can't back up the log on the mirror, but when asked specifically in  about how to shrink an overgrown log file on a mirror database, a  was left that      One way to do this would be to failover to the mirrored database and   do the shrink there. Thoroughly test this out in a non-production   environment to ensure it has the behavior you want/expect.   This seems to suggest that there might be other ways to shrink the log file on a mirror, and this method might not necessarily be safe to do on a production server.  Is there a way to safely shrink the transaction log file on a database mirror? Word or expression for "someone using fake kindness to hide something or his true nature" In Dreamweaver typing &lt;/ will automatically close the last open tag, so you don't have to type it all out, and it helps if you aren't sure if you closed one or not. Is there a plugin that dose this for Notepad++?  This is not like text FX where typing the opening tag/bracket adds the close on the other side of the cursor. This behavior is an auto completion that happens when &lt;/ is typed, DW finds whatever the innermost open tag is, and closes it. I find that, when I have the option of using DW, I will use this feature far more often than anything else that DW provides. Thus I would love to know if there was a way to do it in NPP.  Here is DW preference in question:  Notice how there is a different option that does what text FX does, but I don't want that.  I am not opposed to unorthodox suggestions, some kind of macro, or hack of a plugging. Anything to emulate the functionality. If need be, I could write a plugin, I just have no idea how to go about it. Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $ I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand.  This problem is from some old high school IMO training courses.  I am not sure how to solve it. The only information that may be of help is $$ \sin x=\sum_{n=1}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \quad \cos x=\sum_{n=1}^\infty \frac{ x^{2n}}{(2n)!} \quad \forall \ x $$ Thanks, I am looking for a complete solution, not a description as of what to do. Proving a sequence defined by a recurrence relation converges How can I prove that this iterative sequence converges to $2$? Can I use the convergence definition?  $$a_{n+1} = \frac{a_n}{2} + \frac{2}{a_n},\qquad  a_0 = 4  $$  Thanks for the help. How to reduce the size of VMware disk My Ubuntu image VMware disk has grown to 5GB physical file size. It's a dynamically growing disk with the maximum capacity of 8GB. I cleaned some stuff inside Linux distro and now inside it occupies 3GB. Obviously the physical file size didn't shrink back automatically. Is there a way (if possible free of charge) to shrink this disk back to ~3GB?  I need this to fit the image on the DVD. Footnote citations in biblatex ignore spacing before the citation command; how can I have the same behavior for all footnotes in a document?  Here's a MWE demonstrating the issue:  \documentclass{article} \usepackage[style=verbose-trad2]{biblatex} \addbibresource{biblatex-examples.bib} \begin{document} Spacing is fine before citations     \autocite{kant:kpv}  but not before footnotes    \footnote{A footnote \autocite{kant:ku}}. \end{document} Passing a varible from jQuery to PHP Somewhere in the provided answer:   $$\int \frac{1}{\sqrt{2-x^2}} dx = \sin^{-1}{\frac{x}{\sqrt{2}}}$$  How did they get that? What I have:   $$\frac{1}{\sqrt{2-x^2}} = \frac{1}{\sqrt{2(1-\frac{x^2}{2})}} = \frac{1}{\sqrt{2} \sqrt{1-\frac{x^2}{2}}}$$  $$\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{1-(\frac{x}{\sqrt{2}})^2}} = \frac{1}{\sqrt{2}} \sin^{-1}{\frac{x}{\sqrt{2}}}$$  So I have an extra $\frac{1}{\sqrt{2}}$ ... I probably had some stupid mistakes? It is me who am to blame.    It is I who am to blame.      Which one is correct here? I am bewildered about these two sentences. Java, Collection constructor Finding places to play Magic the Gathering games A friend of mine recently got MTG for the XBox and is talking us into getting real cards.  I'm all up for starting but I dread the idea of playing the same two people over and over with no change epically since one of them is over competitive and exploits every loophole he can to win.  Does anyone know of any a reliable (not just google it) way to find places where people play MTG? If a Windows shop moves "everything" to the cloud, does it still need Active Directory? Taking a spin off of this question:  in a new direction for 2014.  Taking into account a basic Windows infrastructure:   domain controllers Exchange 2007/2010/2013 Sharepoint SQL File Servers / Print Servers AD Integrated DNS AD authenticated 3rd party devices (let's say 802.1X for networking and maybe some content-filtering, etc.) AD/LDAP authenticated "administrative" functions on IT apps/hardware/etc. perhaps some KMS stuff throw in a CA if you'd like home grown apps 3rd party in-house apps   Now, let's rip it all out and decide we are going to the cloud.  We've contracted to move Exchange/Sharepoint/File Services to Office 365.  SQL will now be hosted as well on  something like Azure.  We've gotten away from the need for AD-DNS and simply run everything via a simple Windows DNS server.  We still need 802.1X and would like SSO if possible to our various cloud apps.  Home grown and 3rd party in-house apps would likely stay, but have the ability to use internal user databases instead of AD authentication  The question is...do we really need Active Directory at all?  Or more to the point, AD on-premise or even hosted via Azure or similar (ADFS) or running ADDS on a hosted VM through Azure or similar.  Could/Should we look to something else like a 3rd party SSO option such as  or similar that can provide SSO functionality even if it is as simple as LastPass or similar for each user?  What kind of legitimate needs does AD fulfill if everything else in the cloud?  Could a MS-centric infrastructure get away with not having AD at all if they move everything that previously relied on AD to SaaS offerings that didn't rely on AD authentication? Java Strings: "String s = new String("silly");" is that function must be constant under the following conditions I'm talking about complex function   $f$ is analytic function on a region $D$ that include the point $z=0$. for every $n\in N$ such $\frac{1}{n}$ is in $D$ , the function follows this condition $|f(\frac{1}{n})| \leq \dfrac{1}{2^n}$   is that function must be constant ? What is the MIME equivalent of uuencode for sendmail? e.g., I've got files I want to send by sendmail but the receiving mailbox (for posting pictures to my blog) will not accept uuencode files a la   $uuencode file | sendmail user@domain  I'm guessing it'll accept MIME though. So what command can I use instead?