A proof that $\mathbb{Z}[\frac{1 +i\sqrt{19}}{2}]$ is not Euclidean.
The ring $\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain
The actual period of a pendulum at 90°. Looking for the correct formula Do you have access to any scientific experiment which gives the period of a pendulum when the angle is $90^\circ$:     says $T$ varies to about $18\%$ up to $90^\circ,$ so for a seconds pendulum the complete period would be roughly $2.36.$    I d'like to know the ratio at $90^\circ$ with a good approximation, and the span it can vary in relation to air friction, if the bob  is not massive.  Also, I'd like to know the max. speed of the bob at $P_0$ and how it compares with the speed of same ball sliding without friction on a circular surface, would it be the same?      What I am trying to understand here is rather complex, I'll try to explain:  If the block is placed on an air-track or other frictionless surface, the block would acquire a certain speed, momentum and KE in the opposite direction, right? same amount of momentum is lost by the bob and we can find speed loss.    Isn't the same amount of speed lost by the bob when it is attached to a pendulum?  Is this an important factor or may be the most important factor (more relevant than air friction or friction at the pivot) for the increase of the period when the angle is greater than $30^\circ$?  I figured out that this factor can account for about one third of the $18\%$ increase, can you give a more accurate value?   What I am suggesting is that this (loss of horizontal momentum, which is made clear when the bob is not attached to a rod but slides on a block) is the most important factor that makes the actual period increase when the starting point deviates on the vertical direction. Displaying intersecting roads and creating multiple labels for long roads in QGIS? I am new to GIS and I am using QGIS 1.7.3 on Windows.  Can any one tell me how to display a map from line shapefile with these two characteristics:   Roads are displayed with borders at the sides and intersect at crossroads Name label for long roads are duplicated at specified intervals   Here is the link to an image that illustrates my idea:    (The map was taken from underdark's blog)  This is the best I can achieve by drawing lines with two overlapping simple line, one slightly bigger than the other: A geometry question: Let $ABCD$ be a convex quadrilateral. Prove that AD is parallel to BC. Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{AD+BC}{2}$ . Prove that AD is parallel to BC. I assume that AD is not parallel to BC but I can't find any contradiction. Real valued analytic function defined on a connected set is constant How do "pet" Walkers help ward off other Walkers? In both the comics and the show, Michonne first appears with two walkers that she essentially keeps as pets.    The comics sort of half-explain this - as per :     She notices walkers don't attack one another and hacks off the arms and lower jaws of Mike and Terry. She chains them up by their necks to use them as escorts for an escape that was successful.   In , we see that even if they still sort of smell like Walkers, the Walkers can smell/detect the living and will attack.  In ,      We see Andrea effectively making a new "pet" so that she can use to deter the Walkers that she comes across on her trip to the prison. Along the trip, she is shown to come within close range of Walkers, but they give her space when she points the "pets" in the right direction.   Why would the Walkers not just attempt to go around the "pets"? What about them creates an effective protective barrier? $k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}\,$ if $\,\gcd(l,k)=1$ I would like to prove the following equality: $$k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}$$if $\gcd(l,k)=1$. What methods can I use? Thank you for your help. If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$ If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$. Here is my attempt at a proof. It feels very elaborate and I am not sure if it is correct. Can someone please point out any mistakes or places where I may improve. Thanks!  By the definition of continuity at $a$ of $f,g$ we have that $\lim\limits_{x\to a}f(x)=f(a)$ and $\lim\limits_{x\to a}g(x)=g(a)$. Suppose that $f(a)&gt;g(a)$. Then there exists a $\delta&gt;0$ such that $f&gt;g$ for all $x$ satisfying $|x-a|&lt;\delta$. Then for $x$ satisfying $|x-a|&lt;\delta$ we have $h(x)=f(x)$ and $k(x)=g(x)$ and thus $h$ and $k$ are continuous at $a$ because $f$ and $g$ are continuous at $a$. Now if $f(a)&lt;g(a)$ we simply relabel $f=\tilde{g}$ and $g=\tilde{f}$, so that $\tilde{f}(a)&gt;\tilde{g}(a)$ and we apply the previous result to show that again $h$ and $k$ are continuous at $a$.\  Now if $f(a)=g(a)$ then we can distinguish three cases. $f\geq g$ in a small neighborhood about $a$, $f\leq g$  in a small neighborhood about $a$ or $f\geq g$ on one side of $a$ and $f\leq g$ on the other side. In the first case we assume that $f\geq g$ in some small neighborhood about $a$. Then, since $f(a)=g(a)$ we have $h(x) = f(x)$ in this neighborhood and $k(x) = g(x)$ and again we see that $h$ and $k$ are continuous at $a$ due to the continuity of $f$ and $g$ at $a$. Similarly if $f\leq g$ in a neighborhood around $a$ then $h(x)=g(x)$ and $k(x)=f(x)$ in this neighborhood and $h$ and $k$ are continuous at $a$.   Suppose that just to the left of $a$ we have $f\geq g$ and just to the right of $a$ we have $f\leq g$. Then $\lim\limits_{x\to a^-}h(x)=\lim\limits_{x\to a^-}f(x)=f(a)=h(a)=g(a)=\lim\limits_{x\to a^+}g(x)=\lim\limits_{x\to a^+}h(x)$. Similarly $\lim\limits_{x\to a^-}k(x)=\lim\limits_{x\to a^-}g(x)=g(a)=k(a)=f(a)=\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a^+}k(x)$. So $\lim\limits_{x\to a}h(x)=h(a)$ and $\lim\limits_{x\to a}k(x)=k(a)$ which means that $h$ and $k$ are continuous at $a$. Lastly if $f\leq g$ just to the left of $a$ and $f\geq g$ just to the right of $a$ we can relabel $f=\tilde{g}$ and $g=\tilde{f}$ and apply the last result to show $h$ and $k$ are continuous at $a$.   Again, please point out any mistakes I may have made. Thanks!! Calculating $\lim_{x\to+\infty}(\sqrt{x^2-3x}-x)$ Let $f(x) = \sqrt{x^2-3x}$ and $g(x) = x$. Calculate the following limits, showing all working.  I've done the first two -   $$\lim_{x\to0}f(x)\\ =\lim_{x\to0}\sqrt{x^2-3x}=0$$  $$\lim_{x\to-\infty}\frac{f(x)}{g(x)}\\ =\lim_{x\to-\infty}\frac{\sqrt{x^2-3x}}{x}=\sqrt{1-\frac{3}{x}}=1$$  How do I calculate this one?  $$\lim_{x\to+\infty}(f(x)-g(x))$$ $_POST vs. $_SERVER['REQUEST_METHOD'] == 'POST' Some guy called one of my Snipplr submissions "crap" because I used if ($_SERVER['REQUEST_METHOD'] == 'POST') instead of if ($_POST)  Checking the request method seems more correct to me because that's what I really want to do. Is there some operational difference between the two or is this just a code clarity issue? Is the vector cross product only defined for 3D? Flag pop-up hidden beneath answer box (and it's still transparent!) The flag window on (a comment to) the last question is hidden beneath the answer box, as shown in the following screenshot:    Linked: , which is currently  - this is not fixed for flags on comments! We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or to the indeterminate product $ 0 \cdot \infty $?  Furthermore, why can’t we call $ \infty + \infty $ and $ \infty \cdot \infty $ indeterminate forms?  I’m new to calculus, so please clear up my concepts if you can. Thanks! java.lang.NullPointerException at org.glassfish.hk2.classmodel.reflect.impl.TypesImpl.getType How do I insert a nice tilde that is centered in the middle in a lstlisting? I found another thread about  but I can't apply it because the method described there only works outside of lstlisting as lstlisting displays all the text literally. Any ideas?  Here is how the tilde looks in my document at the moment - it's at the very top of the font and I would like it to be in the middle (like ~ this).    Here is the LaTeX for this fragment:  \section{Remove duplicate, consecutive lines (emulate "uniq")}  \begin{lstlisting} awk 'a !~ $0; { a = $0 }' \end{lstlisting}  Variables in Awk don't need to be initialized or declared before they are being used. What are Harry, Ginny, Ron and Hermione's professions? Has Java 3D changed much since 2005? I'm thinking of picking up the book called "Killer Game Programming" by Andrew Davidson, but its pretty old, now. I know Java, and I've been messing with jMonkeyEngine for a few days, but I'm curious to see if I would benefit from the 3D chapters of this book.   The book was published was in 2005. Has Java changed to the point where this book would now be obsolete, assuming it was up to date at time of publication? How to implement a single instance Java application? Sometime I see many application such as msn, windows media player etc that are single instance applications (when user executes while application is running a new application instance will not created).  In C#, I use Mutex class for this but I don't know how to do this in Java. I have float raster and now I want to convert it to vector. How is it possible with the Python GDAL library?  I have tried with gdal_polygonize.py of GDAL utilities on the command line and it worked excellently. But this utility is based on GDALPolygonize() of C++ library and I want to C++ method GDALFPolygonize() to be used instead, which manipulates float raster data as far as I know. Question: If $(1+x)^n=\binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+\dotsm+\binom{n}{n}x^n$, prove that $\binom{n}{1}-2\binom{n}{2}+3\binom{n}{3}-\dotsm+(-1)^{n-1}n\binom{n}{n}=0$  My attempt:  I wrote the expression for $\binom{n}{k}$ for each term in the L.H.S and then tried to simplify but could not reach the R.H.S.  Please help. I have no idea on how to begin. I am not allowed to use calculus. How to use "`" in the same way it is used on the SE websites?