how do S1 ∈ S2, when R2 is not a subspace of R3?
Why is $\mathbb{R}^2$ not a subspace of $\mathbb{R}^3$?
$a _{1} = \frac{1}{2}, a _{2} = 1, a _{n}= \frac{1}{2} a_{n-1} + \sqrt{a _{n-2} }$  Show by the induction that it's $\le 4$ and also by the induction that $a_{n+1}-a_n \ge 0$  If it comes to first I don't know what to do and if it comes to second I tried many different ways but I couldn't prove it. Converting string of 1s and 0s into binary value I'm trying to convert an incoming sting of 1s and 0s from stdin into their respective binary values (where a string such as "11110111" would be converted to 0xF7). This seems pretty trivial but I don't want to reinvent the wheel so I'm wondering if there's anything in the C/C++ standard libs that can already perform such an operation? I have a chromebook series 5. I want to install Ubuntu onto a USB stick. When I downloaded Ubuntu onto my USB stick, it won't start up.  Do I first have to download Ubuntu onto my chromebook hard drive, open Ubuntu and then make USB stick bootable? And then return back to chrome?  Or is there a way to download directly onto USB and make the stick bootable?  I appreciate any help. Proof of infinitely many primes, clarification Books on cosmology I have finished Carroll's general relativity book, Spacetime and Geometry. I am specifically interested in cosmology, so is there any book which goes more in depth into cosmology? I prefer a mathematically rigorous book, which goes beyond the cosmology chapter in Carroll's book. The Big Bang and inflation theory should be dealt with in depth. How do I turn off the Minecraft 1.7.2 Super Secret Settings? How do I turn off the Minecraft 1.7.2 super secret settings? I tried F3 and F4, hitting Esc repeatedly, all things mentioned on blogs – nothing is working. Fast way to find period-n points of a tent map? I need to find a period-3 orbit for a tent map defined by $$T(x)=\begin{cases}2x &amp; 0\le x\le0.5 \\[2ex] 2-2x &amp;0.5 &lt; x \le 1 \end{cases}$$  So, what is a fast and efficient way of finding it? I only know how to do it by guessing-and-checking. For example, I picked a random point, $x = \frac{2}{7}$ and found that $$T\left(\frac{2}{7}\right)=\frac{4}{7}\\[4ex]T\left(\frac{4}{7}\right)=\frac{6}{7}\\[4ex]T\left(\frac{6}{7}\right)=\frac{2}{7}$$ which is a period-3 orbit. Is there any better way of finding a period-3 orbit , more specifically, a period-3 point? Why am I getting a NoClassDefFoundError in Java? I am getting a NoClassDefFoundError when I run my Java application.  What is typically the cause of this? The man page for GNU find states:    -exec command ;     [...] The  string `{}'  is  replaced  by the current      file name being processed everywhere it occurs in the      arguments to the command, not just in arguments where      it is alone, as in some  versions  of  find.     Both  of  these  constructions might need to be escaped      (with a `\') or quoted to protect them from expansion      by the shell.     That's from the man to find (GNU findutils) 4.4.2.   Now I tested this with bash and dash, and both don't need to have the {} being masked. Here is a simple test:  find /etc -name "hosts" -exec md5sum {} \;    Is there a shell, for which I really need to mask the braces? Note, that it doesn't depend upon whether the file found contains a blank (invoked from bash):   find ~ -maxdepth 1 -type d -name "U*" -exec ls -d {} \;  /home/stefan/Ubuntu One   This changes if the found file is passed to a subshell:  find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d {}' \;  ls: cannot access /home/stefan/Ubuntu: No such file or directory ls: cannot access One: No such file or directory   which can be solved by:  find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d "$0"' {} \;   in contrast to:   find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d "{}"' \;  /home/stefan/Ubuntu One   but that's not what the man page is talking about, is it? So which shell treats {} in a different way? My son has been playing Minecraft on our home computer for a couple of years now.  He just got an iPad Mini 2 for his birthday. Is it possible to transfer his game and his saves onto his iPad Mini 2? How can I get Google to stop thinking I'm a Bot? So, I'm getting the occasional CAPTCHA message, which is irritating but fine, but occasionally Google just blocks my network wholesale. I've submitted automated requests to unblock it, but I have no idea who I can email directly to clarify the issue.  I've run all manner of anti-virus, parsed my router's logs, combed through Fiddler to see if any errant packets were slipping by. Nothing. The only Google searches that happen are made by me through the web-browser.  Who or what do I have to do to get Google to stop falsely flagging my network as a spam-bot? Steps for going from Polyakov action to low-energy effective action (SUGRA?) in String Theory A lot of string-theory questions have been asked here. This one is, hopefully, different in that this inquiry is specific and the expected answer would be more mathematical than philosophical in character.  To give some context, I'm familiar with LQG, QFT etc. I'm a beginner in CFT and string-theory. On reading papers on AdS/CFT and String Theory the following sentence often appears: "The low-energy effective action for $x$ dimensional ST of type $y$ reduces to that of supergravity (SUGRA) in $z$ dimensions with matter fields of types $a$, $b$ and $c$".  The $64,000 dollar question any noob, such as me, wants answered is what are the mathematical/technical steps or arguments which lead us from the fundamental action of string-theory (the Polyakov/Nambu-Goto action) to a low-energy effective description. Of course, I might be asking a question for which the only concise answer is: "go read GSW". Here's hoping otherwise ;) How do I get horizontal lines to the left and right of a heading in Microsoft word? In Word 2007 I want to put a heading in the center of a line with a horizontal line to the left and to the right of the text.  The line must not cross through the text. I have tried using autoshapes and lines - but then the lines move each time I press Enter or &larr; Backspace and I have to put them in place again manually.  At Borders and Shading there is an "Insert horizontal line" option, but this gives me a solid line from the left to the right of the page and I cannot insert text anywhere on it or in it.    The final product should look something like this:     -------------------------  Heading ------------------------- block certain dangerous commands? In my  High school there are few computer having Ubuntu 14.04 installed and many students know the Root  password, how  can I block/disable  certain dangerous  commands like :- rm -rf / rm -rf * :(){:|:&amp;};: to avoid Data loss ? Frequently loaded packages: Differences between pdfLaTeX and XeLaTeX Until now, I have been compiling my LaTeX documents with pdfLaTeX. My standard document preamble includes the "usual suspects" for non-English speakers (, , ) and a bunch of other packages. Assuming I were to convert to XeLaTeX, what modifications of my preamble are advisable? I'm aware that the expansion=true option of the  package is not (yet) available for XeLaTeX, and that  is sort of a default package for OpenType fonts. What other packages (and package options) should be removed and, vice versa, incorporated when switching from pdfLaTeX to XeLaTeX? Number of solutions of an arithmetic function's equation Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation}  where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of distinct prime factors of $x$.  I would like to know if this equation holds for finitely many values of $x$ or infinitely many.  If one uses asymptotic formula (from number theory) it can be written that \begin{equation} \omega(x) \sim \ln x / \ln \ln(x) \end{equation} (from )  and  \begin{equation} \pi(x) \sim x/\ln(x), \end{equation} so the above equation can be rewritten (after simplification) as \begin{equation} 2x\ln(2)=\ln(x)\ln\ln(x)\ln(2x) +f(x) \end{equation}  where $f(x)$ is an error term.  If  I plot left side (which is linear, supposed to be a straight line) and right side(a curve),assuming $ f(x) $ is 100% accurate, then the equation implies(?!) that the right will intersect left side. Since line can intersect a curve ($ln  $ function) finite times, so there are finite intersection point. So, can I say that there are only finitely many values of $x$?  graph example-   here f(x) is assumed as x + sinx  without the f(x)- Disk space usage doesn't add up with df & du The following problem was on a math competition that I participated in at my school about a month ago:      Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.   I will outline my proof below.  I think it has some holes.  My approach to the problem was to say that the following equations must have real solution(s) if the above equation has solution(s):  $$ \cos^2(\sin x)=\sin^2(\cos x)\\ 1-\cos^2(\sin x)=1-\sin^2(\cos x)\\ \sin^2(\sin x)=\cos^2(\cos x)\\ \sin(\sin x)=\pm\cos(\cos x)\\ $$  I then proceeded to split into cases and use the identity $\cos t = \sin(\frac{\pi}{2} \pm t\pm y2\pi)$ to get   $$ \sin x=\frac{\pi}{2} \pm \cos x\pm y2\pi\\ $$  and the identity $-\cos t = \sin(-\frac{\pi}{2}\pm t\pm y2\pi) $ to get   $$ \sin x=- \frac{\pi}{2}\pm \cos x \pm y2\pi.\\ $$  where $y$ is any integer.  I argued that $y=0$ was the only value of $y$ that made any sense (since the values of sine and cosine remain between $-1$ and $1$).  Therefore, the above equations become   $$ \sin x=\frac{\pi}{2} \pm \cos x\implies \sin x \pm \cos x=\frac{\pi}{2}\\ $$  and   $$ \sin x=- \frac{\pi}{2}\pm \cos x\implies \cos x\pm\sin x= \frac{\pi}{2}.\\ $$  Then, by a short optimization argument, I showed that these last two equations have no real solutions.  First, does this proof make sense?  Second, if my proof makes sense, then I feel that it was not very elegant nor simple.  Is my approach the best, or is there a better (i.e., more elegant, shorter, simpler) proof? Can non-magical persons see the Hogwarts Express during its journey to and from Hogwarts? Does the following limit exist? $$\lim_{x \to 0} x\sin\left(\frac{1}{x}\right)$$  If yes, then I have to find it; if no, then I need to give reason why. I couldn’t figure out how to prove this formally, can someone please help me out.