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How could I define function with function such as Coefficient? What is the solution of $\cos(x)=x$? There is an unique solution with $x$ being approximately $0.739085$. But is there also a closed form solution? Notation: Why write the differential first? From reading answers here, I've noticed that some people write integrals as $\int dx \; f(x)$, while other people write them as $\int f(x)\;dx$.  I realize that there is no mathematical difference between the two notation forms, but was wondering why some people choose the first method over the second.  Is there some place in higher maths that it becomes beneficial to write the differential first?  (I, personally, have always used the second method, just because I was taught that way...) How do you use plural or singular form of an adjective? To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$ if $a,b,c$ are non zero positive reals prove $$\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$$ I have used A.M  G.M  inequality as follows:  $$a^3+b^3+c^3 \ge 3abc$$ adding $abc$ both sides we get  $$a^3+b^3+abc \ge 4abc-c^3=c(4ab-c^2)$$ so  $$\frac{1}{a^3+b^3+abc} \le \frac{1}{c(4ab-c^2)}$$ But since $a,b,c$ are positive reals  $$4abc-c^3 &lt; 4abc$$ so  $$\frac{1}{4abc-c^3} \gt \frac{1}{4abc}$$  but i am unable to proceed here Changing my iTunes account details How to check the maximum number of allowed connections to an Oracle database? I need to do an  request in JavaScript. What's the best way to do that?  I need to do this in a Mac OS X dashcode widget. How to write a function (piecewise) with bracket outside? How to write this Piecewise function using LaTeX?  I tried $ \begin{array}{cc}   \{ &amp;      \begin{array}{cc}       0 &amp; x\leq 0 \\       \frac{100-x}{100} &amp; 0\leq x\leq 100 \\       0 &amp; 100\leq x     \end{array} \end{array} $  but it didn't work perfectly, the bracket &quot;{&quot; looks small, see: How do you split a list into evenly sized chunks? Background not changing using gsettings from cron I'm trying to change my wallpaper to a random image using Indrajith Indraprastham's suggested script here:    When I run the script from a terminal window, the bg changes just fine, but when it's run from cron, I'm mailed this error:     (process:21901): dconf-WARNING **: failed to commit changes to dconf: Error spawning command line 'dbus-launch --autolaunch=00216c114dcf433c9bb9009985d607d6 --binary-syntax --close-stderr': Child process exited with code 1   I would appreciate any suggestions. Is $\mathcal{O}_{\mathbb{Q}(\sqrt{5})} = \mathbb{Z}[\phi]$, where $\phi={1+\sqrt{5}\over 2}$ is the golden ratio?  I know that $5 \equiv 1 \mod 4$, so that then $\mathbb{Z}[\sqrt{5}]$ is not closed as far as integers go. But I'm a little confused cause  $\mathcal{O}_{\mathbb{Q}(\sqrt{-3})} = \mathbb{Z}\left[-\frac{1}{2} + \frac{\sqrt{-3}}{2}\right]$? So how do I know when to use $\frac{1}{2}$ rather than $-\frac{1}{2}$? Which one applies for $\mathcal{O}_{\mathbb{Q}(\sqrt{5})}$? How can I use the instant messenger program Kik - most commonly installed on smartphones and other mobile devices - on a laptop or desktop running Win7Pro, without a phone being involved and without needing any kind of Google account?  The two ways of using Kik from my laptop that I have so far found are both inadequate, or at least I do not how to tweak them to make them meet the requirements.  The first is to install on my laptop the Bluestacks emulator of Google's Android operating system, and then to run Kik on it. Unfortunately to run Kik on Bluestacks requires (or at least I have not found a way to avoid) having a Google Play account.  The second is to run Bluestacks on Manymo, the online emulator of Google's Android OS. Unfortunately, using Kik on Manymo similarly requires a Google Play account.  I realise Kik was made for use on a mobile OS, but it is supposed to be cross-platform where mobile OSes are concerned, and therefore not to be tied into Google's mobile OS - or, as far as I know, into any other software supplied by Google or any procedure prescribed by that company, such as giving it personal information - so I am optimistic that someone will be able to advise on what I am asking for here.  I notice that Microsoft have made a , so there might be a way to get Kik to work on this without dealing with Google. I couldn't find anything online about this, though. Let $f:G\to K$ be a surjective group homomorphism and $H$ be a subgroup of $G$ satisfying $N_G(H)=H$. Then $N_K(f(H))=f(H)$. I'm trying to prove or disprove it. perhaps, this proposition is true. But, how can we prove it? Edit: I thought it is true because WIKI said it is true without proof Why aren't my guide and other dead NPCs respawning in Terraria? I am currently in hard mode and during a blood moon, several of my NPCs including my guide and my healer died. it's been several in game "days" now and they have yet to respawn. Is there a way to check why they aren't respawning?   Is there a list of reasons for the NPCs to not respawn?   Edit in reponse to lunbok's answer:  1. They don't have appropriate housing - There is an in-game housing query function (changes your cursor into a ?) that lets you click on your house to see if it is still a suitable house and I clicked in every room in my house and they are all listed as "This housing is suitable" or "This housing is already occupied".  2. Their spawn conditions aren't met anymore. - This doesn't explain why my guide isn't respawning since they don't need a condition to respawn (other than having a suitable housing which the point above already prove isn't the problem).  3. There's a goblin invasion going on. - Do the invasions last that long? It has been about two reals days and a lot more in-game "days" now. As far as I can tell there shouldn't be an invasion going on.  4. It could be because you built your house in a part of the world that is now corrupted I've just made a set of armor by using a male body mesh model I downloaded and selecting certain faces from the model and duplicating them to create viable armor pieces.   Since the pieces are just duplicated from the same mesh male model, then it's still part of the same object.   How do I separate the armor piece from the model to make the pieces separate? Split string every nth character? I found this question in a book.   How many terms of the Maclaurin expansion of $(1+x)^{82}$ are needed to guarantee finding a value of $1.1^{82}$ to an accuracy of $10^{-6}$?  This is how I tried to do it:  We want the Lagrange error to be less than $10^{-6}$:  $$R_{n}(0.1)&lt;10^{-6}$$  Now to find $R_{n}(0.1)$:  $$R_{n}(0.1)=\frac{f^{(n+1)}(0.1)\cdot0.1^{n+1}}{(n+1)!}$$  After writing the values of ${f^{(n+1)}}(x)$, I find a pattern and say that  $${f^{(n+1)}}(x)=\frac{82!}{(82-n)!}(1+x)^{82-n}$$  $${f^{(n+1)}}(0.1)=\frac{82!}{(82-n)!}(1.1)^{82-n}=\frac{82!(1.1)^{82-n}}{(82-n)!}$$  Now just substitute all of this back to $R_{n}(0.1)&lt;10^{-6}$:  $$\frac{\frac{82!(1.1)^{82-n}}{(82-n)!}\cdot0.1^{n+1}}{(n+1)!}&lt;10^{-6}$$  Great! Now I have an inequality that I can solve. I can't graph this so I just use trial and error (which seems like quite an intensive way to do it, so if anyone has a better way I'd love to know).  $$\frac{\frac{82!(1.1)^{82-n}}{(82-n)!}\cdot0.1^{n+1}}{(n+1)!}-10^{-6}&lt;0$$  This seems to work first for $n=28$. That means the highest power of $x$ that I need is 28, so the number of terms in the expansion needs to be $n+1=29$. But the book's answer is       30   What am I doing wrong here? I'm still quite new to the power series expansions so if there's anything unclear in my working, please let me know (also, as a side note, the book itself may be wrong, though that isn't very likely I'd assume). Does a constantly accelerating charged particle emit EM radiation or not? Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$ This is a follow-up to  question. I came across a tighter potential bound and checked it numerically for $0\le x\le 5$. I think it holds for all positive $x$, can anyone see a proof?  $$1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$$  Note: using analysis for previous  you can show that $1-\exp(-k x^2)$ is an upper bound on $\text{erf}(x)^2$ for $k=2$ and a lower bound when $k=1$. The factor $k=4/\pi$ comes out when numerically searching for tightest upper bound. Not only does it seem to give an upper bound, but it also tracks $\text{erf}(x)^2$ very closely.  Dashed graph below is $\text{erf}(x)^2$, red is $1-\exp(-k x^2)$ for $k=4/\pi$, other two graphs are for $k=1$ and $k=2$