Find the value of $\sqrt3\csc 20^o -\sec 20^o$
Can $\frac{\sqrt{3}}{\sin20^{\circ}}-\frac{1}{\cos20^{\circ}}$ have two values?
Series of books featuring dragons, enchanted forests, magic swords, etc When I was younger I read a series of 3 or 4 books involving largely traditional mages/dragons/enchanted forests.  I'd like to buy it for some of my younger extended family members but I can't remember the name of it.  Here's what I remember from this:   I read the last book first, in which the son of a woman turned out to be largely one of the main characters in the other books takes an enchanted sword into a forest on a quest to do something. At one point in one of the earlier books the group was travelling through a swamp like area to get to a tower where a witch(or mage lived). There was a king (possibly the son's father) who was frozen in a castle during the last book. The evil mages/witches would melt (but not die) when some spell was cast on them. I read the series circa 1992 (estimated), comparable to the DragonLance series it was geared towards a younger audience (I don't remember any sex/brutal slayings of anyone). I believe in the first book the mother of the son in the last book runs away from home and becomes something of a maid to a dragon. I can't remember if there were evil dragons or not, I seem to remember that at least some of them were good (or at least largely neutral).   If I think of any other details I'll add them. Change the size/width of tabs in Notepad++ Roots of unity filter, identity involving $\sum_{k \ge 0} \binom{n}{3k}$ How do I see that$$\sum_{k \ge 0} \binom{n}{3k} = (1 + 1)^n + (\omega + 1)^n + (\omega^2 + 1)^n,$$where $\omega = \text{exp}\left({2\over3}\pi i\right)$? What is the underlying intuition behind this equality? What content type to force download of text response? I need to stream a file to the Response for saving on the end user's machine.  The file is plain text, so what content type can I use to prevent the text being displayed in the browser? Find the default application for a certain extension Given xdg-open and an extension, is there a way to get the application which xdg-open is set to for that particular extension?  For example given xdg-open and .jpg the result is eog. How to embed fonts in HTML? I'm trying to figure out a decent solution (especially from the SEO side) for embedding fonts in web pages. So far I have seen , which doesn't even work on Firefox, and . The second solution is for titles only. Is there a solution available for full text? I'm tired of the standard fonts for web pages.  Thanks! Fullscreen YouTube video crash I have browsed several ASIC manufacturer's webs, but I haven't found an actual number. I assume there would be a fixed cost associated with creating masks and such and then there will be a cost per unit.     Note: that I don't actually want to have an ASIC made, I'm just curious. Memory and CPU optimizers: Are they gimmicks? character "5" repeated in text fields, and system freezes I am running Ubuntu 15.10, with the latest updates.  When I type in a text field (google search, type name of file, etc..) the number 5 gets typed on its own and it keeps repeating until I press ESC and when I try to continue typing the number 5 resumes filling in the text field.  I have tried to set the keyboard repeat to zero, but does not help.  This does not happen all the time, sometimes if I shut down the computer it then behaves normally when I restart it, other times the computer becomes useless because I cannot type in any of the text fields.  Thanks Sal Novel about interstellar war: Both sides win, both sides think they lost I recall reading this novel in the &rsquo;90s; my recollection is that the author was better-known for bad fantasy than for SF (but the SF of his that I&rsquo;d read was better than his fantasy).  The story opened in media res just before a battle was joined; there is a dialogue between the captain of one of the ships and one of his crew. We never see the actual battle; we instead see the conquest of Earth by the aliens - and the conquest of the alien homeworld by Earth.  It is slowly revealed that the two sides contrived a way to both avoid the destruction of actual war, and solve their respective worlds&rsquo; respective problems of overpopulation, infrastructure decay, et multae ceterae. We also see that the solution was that the conquering aliens were secretly selecting Earthmen to rule their homeworld, while the humans were selecting aliens to rule Earth.  About halfway through, there is suddenly a third world that must be &lsquo;conquered&rsquo; and brought into the network. The plot gets complicated after that. How to re-enable tray icons for applications in Ubuntu? Matrix Ring of a Semisimple Ring I recently read the concept of semi-simplicity of a (not necessarily commutative) ring.  A ring $R$ is said to be semi-simple if $R$ as a left module over itself is a semi-simple module (This in turn means that each submodule (i.e, left ideal) of $R$ is a direct summand for $R$.)  I am grappling with the following question:     If $R$ is semi-simple, then so is the matrix ring $M_n(R)$.   It is a theorem that each left module of a semi-simple ring is a semi-simple module. Thus $M_n(R)$ is a semi-simple module of $R$. But this doesn't help me.  I need to show that each left ideal of $M_n(R)$ is a direct summand for $M_n(R)$. Is there a characterization  of the left ideals of $M_n(R)$ in terms of the left ideals of $R$?  Lastly, does the converse also hold?     If $M_n(R)$ is a semi-simple ring, then is $R$ necessarily a semi-simple ring?   Thanks. Can I remove a digital signature from a DLL? Interview: How to ensure that a thread runs after another? How do I derive the Lorentz contraction from the invariant interval? Reviewing some basic special relativity, and I stumbled upon this problem:  From the definition of the proper time: $$c^2d\tau^2=c^2dt^2-dx^2$$ I was able to derive the time dilation formula by using $x=vt$: $$c^2d\tau^2=c^2dt^2-v^2dt^2=c^2dt^2\left(1-\frac{v^2}{c^2}\right)\rightarrow d\tau = dt\sqrt{1-\frac{v^2}{c^2}}=t/\gamma$$  Now, I would very much like to be able to derive the length contraction formula in a similar fashion, and feel strongly that this should be possible. The definition of the invariant interval is: $$ds^2=dx^2-c^2dt^2$$ using $t=\frac{x}{v}$ I tried: $$ds^2=dx^2-\frac{c^2}{v^2}dx^2=dx^2\left(1-\frac{c^2}{v^2}\right)\rightarrow ds=dx\sqrt{1-\frac{c^2}{v^2}}$$  This is where I'm stuck: I don't see how this can be converted to a Lorentz factor...  Any help that will allow me to arrive at the desired result $ds=\gamma dx$ would be very much appreciated. Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.  One series that popped into our discussion was $\sum\limits_{n=1}^{\infty} \frac{1}{n^n}$.  Proving the convergence of this series is trivial but finding the value to which converges has defied me so far. Mathematica says this series converges to $\approx 1.29129$.  I tried Googling about this series and found very little information about this series (which is actually surprising since the series looks cool enough to arise in some context).  We were joking that it should have something to do with $\pi,e,\phi,\gamma$ or at the least it must be a transcendental number :-).  My questions are:   What does this series converge to? Does this series arise in any context and are there interesting trivia to be known about this series?   I am actually slightly puzzled that I have not been able to find much about this series on the Internet. (At least my Google search did not yield any interesting results). If $E$ is a connected space, then so is $\overline{E}$. I have to prove that if $E$ is a connected space, then so is $\overline{E}$ (the closure of $E$) a connected space. I tried to prove the contrapositive. So suppose that $\overline{E}$ is not connected in a metric space $X$. This implies there are $A\subset X$ and $B\subset X$ such that $A\cap B=\emptyset$, $A\cap \overline{E}\neq \emptyset$, $B\cap \overline{E}\neq \emptyset$ and $\overline{E}\subset A\cup B$. I will try to prove that this $A$ and $B$ will also work for $E$. Suppose that $A\cap E=\emptyset$ and $B\cap E=\emptyset$, this implies that $A\cup B$ is a subset of the limit points of $E$, because $A\cap \overline{E}\neq \emptyset$ and $B\cap \overline{E}\neq \emptyset$. This contradicts the fact that $\overline{E}\subset A\cup B$, so we have to conclude that $A\cap E\neq \emptyset\neq B\cap E$. Since $E\subset\overline{E}\subset A\cup B$, we have that $E$ is also not connected.  My question is whether my proof is correct, because I have a little bit doubt. If it's not correct, what is the best I can do? Thanks in advance!   Edit1: The definition we use for a connected space $E$ in a metric space $X$ is that if $E$ is connected, then there are no open $A,B\subset X$ such that $A\cap B=\emptyset$, $A\cap E\neq \emptyset\neq B\cap E$ and $E\subset A\cup B$.  Edit2: $A$ and $B$ must be open. Using Windows 10 in a virtual machine. Can it be done? I am currently running 8.1. I'd like to try out Windows 10 and if I like it upgrade. If not, I'll keep running as I am.   I've never used a virtual machine before: can Window 10 be used from a virtual machine as a "sandbox", and if so how? Alternatives to "and/or"?