Multilinestring to shapefile using QGIS?
Converting WKT to SHP?
A characterisation of quadratic extensions contained in cyclic extensions of degree 4 Let $D$ be a squarefree integer. I am trying to prove that $\mathbb Q[\sqrt D]$ is contained in a Galois extension of $\mathbb Q$ with Galois group $\mathbb Z/4$ if and only if $D$ is the sum of two squares, $D = a^2 + b^2$ with $a,b\in \mathbb Q$.   The hint in the exercise in Dummit and Foote suggests considering the extension $\mathbb Q[\sqrt{s + s\sqrt D}]$ for the forward direction. However, I am unsure how to show that this extension is Galois, much less that its Galois group is cyclic of order 4.   For the reverse direction, I expressed the given Galois extension as $\mathbb Q[\sqrt{a+b\sqrt D}]$ for some $a,b\in \mathbb Q$. I showed that this extension has $\mathbb Q[\sqrt{c^2 - d^2 D}]$ as a subfield for some $c,d \in \mathbb Q$. From this, I deduced that the $\sqrt D$ and $\sqrt{c^2 - d^2 D}$ generate the same extension, so $\sqrt D = x \sqrt{c^2 - d^2 D}$ for some $x$, and hence that $D = \frac{x^2 c^2}{1 + x^2 d^2} $. It seems that there should be a way to decompose this as a sum of two squares, but I did not see how.   I did not tag this as homework as I am studying it independently. Material will not assign to selected vertices A little bird told me that if $2^n+1$ is prime, then $n$ is a power of $2$. I tend not to trust talking birds, so I'm trying to verify that statement independently.  Suppose $n$ is not a power of $2$. Then $n = a \cdot 2^m$ for some $a$ not a power of $2$ and some integer $m$. This gives $2^n+1 = 2^{a \cdot 2^m}+1$. Now I suspect there's a way to factor that, but I don't see how. Can someone give me a hint? Run as a different user on a shortcut How do I run as a different user on a shortcut in Windows 7?   On Windows XP, I had the ability to mark a shortcut as being run by a different user, so that every time I ran it it would prompt me for a username/password. This let me have two shortcuts for things like , one for my normal account and one for my domain administrator account which has access to production servers.  I can get to the 'Run as different user' option with Shift + right-click, but I can't see an option anywhere that would let me mark the shortcut as doing this every time. Every separable Banach space is isomorphic to $\ell_1/A$ for some closed $A\subset \ell_1$ How to prove the following mind-blowing fact?     Let $X$ be a separable Banach space and let $\ell_1$ be the space of all absolutely summable scalar sequences. Then there exists such closed subspace $A\subset \ell_1$ that factor space $\ell_1/A$ and $X$ are isomorphic as normed spaces.   Edit: So what, this is like a classification up to isomorphism of all separable Banach spaces? Each separable Banach space corresponds to some closed subspace of $\ell_1$? Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law] Are we slightly lighter during the day and slightly heavier at night, owing to the force of the Sun's gravity? Using $g = \frac{Gm}{r^2}$, the force on a point mass located at 1 AU from the Sun ($m = 2 \cdot 10^{30} \text{ kg}$) is about ~0.006 N/kg.  Does that mean that, e.g., a 70 kg person is ~42g lighter during the day, and ~42g heavier at night? That seems like it could make a big difference for things like measuring gold bars or other weight-sensitive items. (Gold arbitrage: buy your gold during the day and sell it during the night! Risk-free profit!)  This makes me suspect that I'm overlooking something obvious, because a ~0.05% weight difference seems like something everyone would have noticed long ago. So what am I missing?    Edit: A few answers below indicate that there shouldn't be any weight difference, because the Earth orbits the Sun in free-fall. But if that's the reason, does that mean that a 1:1 tidally-locked Earth-Sun system wouldn't experience differential gravity from the Sun on opposite sides? That doesn't seem right. This question has troubled me for ages despite my several attempts of looking it up in dictionaries or usage books. Do we say, "Do you have any ideas" or "Do you have any idea"? I do see an example where "any" means "it doesn't matter who/which/what", therefore "You can borrow any book you like." Also, does it matter between using it in questions and negatives? In ArcGIS there is a Erase Point tool how can this be done in R The new hot questions algorithm sometimes recurses The hot questions algorithm . I like the effects, since I see lots of new interesting stuff every day rather than waiting 6-8 weeks varying amounts of time for the list to update.  However, the new algorithm also has the interesting side effect of recursing the same question:    Of course, this is a world-shattering workflow-breaking unicorn-hating waffle-despising ERROR tremendously minor matter. But, if the SE devs have opportunity during their amazingly large amounts of free time, it might be nice to add a check for the current page before populating the hot questions list. Direction of Friction on two stacked boxes How to combine two jQuery results How do you combine two jQuery search results? eg:  var $allFoos = $('.foo'),     $allBars = $('.bar')     $allFoosAndBars = $allFoos + $allBars;   Obviously, I just made up that last line, but I hope it makes it sorta clear what I mean. To be clear, the example is greatly simplified, and it could be any arbitrary sets i'm talking about, so $('.foo, .bar') is not what I'm after. $\lim_{n\to\infty}a_nb_n=? $ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=\infty$ A. $\lim_{n\to\infty}a_nb_n=?$ given that $\lim_{n\to\infty}a_n=0$ and $ \lim_{n\to\infty}b_n=\infty$  B. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=0$  C. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=\infty$ and $\lim_{n\to\infty}b_n= \infty$    These seem to be difficult situations I come across when I have to evaluate limits.   Some examples,    $\lim_{n\to\infty}n^a\sin(1/n^b)$ with $a,b&gt;1$   $\lim_{n\to\infty}{n^a\over \sin(1/n^b)}$ with $a,b&gt;1$   $\lim_{n\to\infty}{1/n^a\over \sin(1/n^b)}$ with $a,b&gt;1$   $\lim_{n\to\infty} nf(x_n)$ [where we only know that $\lim \ f(x_n) = 0]$    My first question is : Are the two situations  A and B  equivalent ? I feel that they are equivalent as $\lim_{n\to\infty}b_n=\infty \iff \lim_{n\to\infty}{1\over b_n}=0$, The problem is that I'm not sure if we can apply the product rule for limits to take the limit inside.  My second question is : What are some ways to solve such problems ? I only know that we can apply stolz- cesaro in some cases. Is there a term for an issue or activity that occurs only until the moment that you ask someone else to watch? Is there a term for the coincidence that occurs when an issue ONLY occurs until the moment that you ask someone else to watch or help you resolve it?    For example, having an issue with software until you try to show someone, then it stops being problematic.  Or an issue with the computer/phone that resolves as soon as a help ticket is put in.    The final example is when you have a problem with your car (funny noise, etc) - you hesitate taking it in for weeks because you know it will disappear as soon as the mechanic drives it...then it does.    This is different than a , where the action of applying a debugger, for example, changes the behavior of the activity being watched.  It's also different from a 'works on my machine' scenario, where only the original user has the problem.   What is it called when a problem resolves itself just by the mention of it?  Jinxing the issue? :)  I see this every day supporting my software product -- the users are experienced and are not making mistakes at one time then doing something different when I watch.  The problems also truly resolve (coincidentally) as soon as they ask for help - they do not reoccur.  I guess it's a case of synchronicity? What do the symbols displayed by ls -F mean? I noticed that if I run ls -F on a directory, some of the entries have a * or a @ after them.   spuder@ubuntu:~$ ls -F /sbin acpi_available*   getpcaps*           lvmconf*                 ntfscp*        start-stop-daemon* agetty*           getty*              lvmdiskscan@             ntfslabel*     status@ alsa*             halt@               lvmdump*                 ntfsresize*    stop@ alsactl*          hdparm*             lvmsadc@      spuder@ubuntu:~$ ls -F ~ daq-0.6.1/  examples.desktop       noname-cache.lib  snort-2.9.1/   Templates/ Desktop/    jpgraph-1.27.1/        noname.sch        snortfiles/    Ubuntu One/ Documents/     According to the ls man pages   spuder@ubuntu:~$ man ls ... -F, --classify   append indicator (one of */=&gt;@|) to entries ...   I'm guessing that @ means symbolic link,     What do these other indicators mean ( */=&gt;@| )  ? While learning about Random variables I came across the mean of random variable X.  The definition says that the expected value of random variable E(X) = Mean of Random variable X  I am not able to understand why is that so. Can any one please help me with it Prove inequality $\sqrt{\frac{2a}{b+a}} + \sqrt{\frac{2b}{c+b}} + \sqrt{\frac{2c}{a+c}} \leq 3$ For which $a \in \mathbb{C} -\ \{0,1\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$.  My attempt:  First, we can see that any such solution must satisfy:  $f(z)=f(a^kz)$ for all $k \in \mathbb{N} $.  If $|a|&lt;1$:  The series $z_{k} = a^k$ converges to 0 which is an accumulation point, and $f(z_i)=f(z_j)$ for all $i, j\in \mathbb{N} $. Thus $f$ must be constant.  If $|a|=1$:  For all $a \neq 1$ , $f$ must be constant on any circle around $0$, so again $f$ must be constant.  My quesions are:   Am I correct with my consclusions?   Also, I'm stuck in the case where $|a|&gt;1$. Any ideas?  Thanks Calculate E[X] from incomplete data? The exercise I'm doing describes the random variable $X$ as the following  | Number of cars |  0  |  1 |  2 |  3 |  4+ |  | % of families  |  15 | 45 | 25 | 13 |  2  |   Then it asks me to evaluate $E[X]$. But if there could be cases with 4+ cars, wouldn't that make it impossible to calculate the expectancy? Installing .pkg with terminal ? Is it possible to install a .pkg using a terminal ? (I wanted to install an app through ssh).