How to find the continued fraction of $\sqrt{n}$, for an integer $n$?  I saw a site where they explained it, but it required a calculator. Is it possible to do it without a calculator?
If I want to find the continued fraction of $\sqrt{n}$ how do I know which number to use for $a_0$? Is there a way to do it without using a calculator or anything like that? What's the general algorithm for computing it? I tried to read the wiki article but was overwhelmed and lost. I tried Googling but couldn't find a website that actually explained this question.  If anyone has a good site that answers these questions either, please let me know. Thanks!
Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$ I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some questions from previous exams. One of the questions is the following:           Prove that $\displaystyle{\sum_{n = 0}^{2013} a_n z^n \neq 0}$ if the coefficients  satisfy $a_0 &gt; a_1 &gt; \dots &gt; a_{2013} &gt; 0$ and $|z| \leq 1$.      I tried to approach the problem by using the reverse triangle inequality to try to isolate the leading term, which I thought should be the constant coefficient, as follows:  \begin{array} . \left|  \sum_{n = 0}^{2013} a_n z^n \right| &amp;= \left| a_0 +  \sum_{n = 1}^{2013} a_n z^n \right| \\ &amp;\geq |a_0| - \left|  \sum_{n = 1}^{2013} a_n z^n \right|\\ &amp;\geq |a_0| -  \sum_{n = 1}^{2013} |a_n z^n|\\ &amp;\geq a_0 -  \sum_{n = 1}^{2013} a_n |z^n|\\ &amp;\geq a_0 -  \sum_{n = 1}^{2013} a_0 |z^n|\\ &amp;= a_0\left( 1  -  \sum_{n = 1}^{2013} |z^n| \right) \end{array}  Then for $|z| &lt; 1$, we have that this last expression is equal to  $$ a_0\left( 1  -  |z| \frac{|z|^{2013} - 1}{|z| - 1} \right) $$  and my hope was to prove that this was always positive, nevertheless, after failing to prove it, I plotted the expression inside the parenthesis as a function of $|z|$, and found out that it changes sign in the interval $(0, 1)$.  Question  Can my approach somehow be made to work? And if not, how can I prove that the sum $\displaystyle{\sum_{n = 0}^{2013} a_n z^n \neq 0}$ under the given conditions?  Thank you very much for any help. IF statement inside a LINQ SELECT Why aren't particles constantly "measured" by the whole universe? Let's say we are doing the double slit experiment with electrons. We get an interference pattern, and if we put detectors at slits, then we get two piles pattern because we measure electrons' positions when going through slits. But an electron interacts with other particles in a lot of different ways, e.g. electric field, gravity. Seems like the whole universe is receiving information about the electron's position. Why is it not the case and the electron goes through slits "unmeasured"?  Bonus question: in real experiments do we face the problem of not "shielding" particles from "measurement" good enough and thus getting a mix of both patterns on the screen? Which PHP function can return the current date/time? When I created my account, I used Facebook Connect as my only login type and I can only log in to my account using it.  I wish to create a Stack Exchange login to be able to log in with instead of Facebook Connect, but I can't find how to do this. It seems that the option for creating a Stack Exchange login is only present when creating a new account? $(this) inside of AJAX success not working I need to get a history of which processes have run on my Windows machine, and when they were run.  However, I cannot use any third-party software, as I cannot guarantee that it will always be running.  Is there any way to get this information only using Windows built-in functionality? Destructors of builtin types (int, char etc..) In C++ the following code gives a compiler error:  void destruct1 (int * item) {   item-&gt;~int(); }   This code is nearly the same, I just typedef the int to another type and something magic happens:  typedef int myint;  void destruct2 (myint * item) {   item-&gt;~myint(); }   Why does the second code work? Does an int get a destructor just because it has been typedefed?  In case you wonder why one ever would like to do this: This comes from refactoring C++ code. We're removing the standard heap and replacing it with selfmade pools. This requires us to call placement-new and the destructors. I know that calling destructors for primitive types is useless, but we want them in the code nevertheless in case we later replace PODs with real classes.  Finding out that naked int's don't work but typedefed ones do was quite a surprise.  Btw - I have a solution that involves template-functions. We just typedef inside the template and everything is fine. Creating a temporary directory in Windows? What's the best way to get a temp directory name in Windows?  I see that I can use GetTempPath and GetTempFileName to create a temporary file, but is there any equivalent to the Linux / BSD  function for creating a temporary directory? How should I deal with becoming discouraged as a graduate student? I am a third year (starting fourth year in the fall) PhD student in mathematics. I've passed all qualifying exams and am currently doing research. As far as I can tell, I am not doing poorly. I have the good fortune of having a great advisor, being in a very supportive department, and having friends and family who genuinely care about my success.   The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere. Sometimes, rarely, I do figure something out and that feels wonderful, but the overwhelming majority of my time appears to be spent banging my head against a mostly figurative wall. I am not complaining about the material being hard, and I am not averse to putting in hard work, but I get frequently discouraged when I realize the vast volumes of mathematics that I yet know nothing about (and probably never will). It's very hard to quantify progress - in particular, there are too few tangible returns after too many hours worked. I find myself thinking along the lines of "Oh, if only someone actually smart were thinking about this problem they would have solved it in moments" and so on.   I've talked about this to some number of people; here is some advice I have received:   Take a day off. Putting in hours upon hours of trying things doesn't magically lead to a solution, particularly if the brain is tired and just wants to sleep. Have a hobby. Since math research doesn't exactly provide instant gratification, a hobby might provide some instead.  Talk to other graduate students. Realize that many graduate students go through this.   I'm interested to know how other folks have dealt with being discouraged as a graduate student. Does it get better with time and experience? Is this a sign that research is not for me and that I should seriously consider a life outside academia? Accurately simulating the lots of dice rolls without loops? OK so if your game rolls lots of dice you can just call a random number generator in a loop. But for any set of dice being rolled often enough you will get a distribution curve/histogram. So my question is there a nice simple calculation I can run that will give me a number that fits that distribution?  E.g. 2D6 - Score - % Probability  2 - 2.77%  3 - 5.55%  4 - 8.33%  5 - 11.11%  6 - 13.88%  7 - 16.66%  8 - 13.88%  9 - 11.11%  10 - 8.33%  11 - 5.55%  12 - 2.77%  So knowing the above you could roll a single d100 and work out an accurate 2D6 value. But once we start with 10D6, 50D6, 100D6, 1000D6 this could save a lot of processing time. So there must be a tutorial / method / algorithm that can do this fast? It is probably handy for stock markets, casinos, strategy games, dwarf fortress etc. What if you could simulate the outcomes of complete strategic battle that would take hours to play with a few calls to this function and some basic maths? Find all files that are NOT of a specific type/extension in folder? What has been proved about the big bang, and what has not? Ok so the universe is in constant expansion, that has been proven, right? And that means that it was smaller in the past.. But what's the smallest size we can be sure the universe has ever had?  I just want to know what's the oldest thing we are sure about. Can't kill a sleeping process Intuitively, $\lim\limits_{t\to -\infty}te^t=0$, since $e^t\to 0$ much faster than $t\to -\infty$.   Is there a way to more rigorously compute this? Writing it as $e^t/(1/t)$ seems resistant to l'Hopital's rule. Starters on Stack Overflow tend to lose interest as soon as their first question starts getting downvotes. Their first question also sometimes gets closed because it was either not framed properly or it missed some very common things.  There should be way to cope with this for instance give some starter bonuses to new users. Booting Ubuntu Failure : error: attempt to read or write outside of disk 'hd0' I have installed Ubuntu 12.10 in a Western Digital external hard drive (320GB). This is a complete installation, not a live USB.  When I plug it in my HP desktop I go to the BIOS settings and boot off the hard drive, everything work perfectly as it should. Now this works on every single computer and laptop in my house (all HP), except for ONE. My HP ProBook 4530s.  When I select to boot of the USB I get the message:  error: attempt to read or write outside of disk 'hd0'   Now, I have removed the hard drive from my laptop and the external drive is the ONLY drive plugged in. Below is a screenshot of the message on the screen. After the message I navigate to ls / (as shown below):    After here I try to acces other folders under ls /, for example, I try to go to ls /boot to get to the grub folder. Then I get the same message as before: as shown below:    grub rescue&gt; ls /boot  error: attempt to read or write outside of disk 'hd0' grub rescue&gt; _   The only folders I can access without getting the message again are /home, /run and /usr.   So how do I:   Boot Ubuntu from GRUB2 (this screen) manually Set to automatically boot Ubuntu If possible an explanation for this problem   Thanks! Continuous probability distribution with no first moment but the characteristic function is differentiable I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere.  Cauchy distributions do not fulfill this, as their characteristic functions are not differentiable at $0$. Does anybody here have an example? At least it's my understanding that AES isn't affected by known-plaintext. Is it immune to such an attack, or just resistant? Does this vary for chosen-plaintext? Finding the perimeter of a right triangle with area given