May we write $\sqrt{x^2}=\pm x$.        Is  $\sqrt{x^2}=\sqrt{(\pm x)^2}=\pm{x}$ true ?  But we write $\sqrt{x^2}=|x|=x$  What is the actual logic?
Good-morning Math Exchange (and good evening to some!)  I have a very basic question that is confusing me.   At school I was told that $\sqrt {a^2} = \pm a$  However, does this mean that $\sqrt {a^2} = +2$ *and*$-2$ or does it mean: $\sqrt {a^2} = +2$ *OR*$-2$  Is it wrong to say 'and'? What are the implications of choosing 'and'/'or'  Any help would be greatly appreciated. Thanks in advance and enjoy the rest of your day :)
General question I have an app X installed on my device in version Y. I don't want to update it to any newer version (Y+z) for some reason. Still, the Google Play Store app always lists updates for X, which keeps me from a simple &quot;update all&quot;. My question, put to a single sentence:  How to hide updates for X permanently from the &quot;available updates&quot; list?   Specific examples First, I know there are alternatives as e.g. the , which can handle this quite well (skip this update, skip all future updates). I prefer a &quot;generic solution&quot;, which does not depend on a 3rd party's market app. Second, I know that  includes a &quot;Market doctor&quot;, which can be used to disconnect an app from Google Play.1 However, in my specific case that only worked for a few hours: I disconnected GTalk, as I don't want it replaced by Hangouts. The update notification disappeared for a few hours, then it was back. No idea what restored the &quot;link&quot;. Third, just to make sure: I'm not talking about the &quot;auto update&quot; feature as discussed e.g. in . I was rather hoping for something close to , but without the need to &quot;freeze&quot; the app in question (which I still want to use, but in the version installed, not in a &quot;future version&quot; available). Fourth: No, uninstalling the app in question is no option here, though it would perfectly hide any future updates from being displayed :) Nor is &quot;freezing&quot;, as the app is used regularly. Fifth: There is a related question for my GTalk example:  However, I want the opposite: No update.  It might very well be that &quot;Google Apps&quot; (and other pre-installed apps) are handled in some specific way, and restore their &quot;market links&quot; themselves somehow. For this case, my question includes: How to prevent that from happening?  Update There's a  discussing the very same issue. Until recently, a work-around for &quot;user installed apps&quot; was to uninstall them and then sideload the .apk – but even that stopped working. Well, re-signing the .apk with a different key might work for user-apps. But both would fail for pre-installed apps (can't sideload them to override; different signature would refuse that install completely). Moreover, there's an  on this by some &quot;joel.bou..&quot;. Not hard to guess, though blanked out: that would be Joël Bourquard. If it's really him, then that's really an issue (for those who don't know: Joël is the dev behind Titanium Backup). Let me quote from the issue:  For the record: in the Android Market days it was possible to &quot;Attach&quot; or &quot;Detach&quot; apps (on rooted devices) with a system tool such as Titanium Backup. But nowadays (with the Google Play Store app) this has become nearly impossible to do, because the complete app list seems to sync from Google's servers and the user has no control over that.  Which explains why detaching with apps like Titanium Backup is no longer permanent. And I'm afraid I've asked the impossible. But I thought the same in the past with other questions, and surprisingly got solutions here. So I do not yet give up hope on this one!  1: what that does is explained by eldarerathis  (thanks to  for digging that up!) Basically, I acidentally spawned too many ArmorStands due to a wrong code writing , resulting in the game crashing, and wanted to know how to delete those armorstands from the World save folder. They hava names and I belive they also have a scoreboard set, if that helps.   I have already tried to open the world but it crashes, I tried to open a server (Since it worked the last time I tried it) and killing it all from the console but, different from the last time, it crashed before killing them and now I just tried MCedit but it crashes as well. Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b&gt;0$.   My Try:  So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq 2\pi\}$.   On $\Gamma$, $z=a\cos t+i b \sin t$, so $|z|^2=z\bar{z}=a^2\cos^2 t+b^2 \sin^2 t$.   Now, $dz=-a\sin t+i b \cos t dt$.   Hence, the integral becomes $\displaystyle \int_\Gamma \frac{dz}{z(iab+(\sin t \cos t)(b^2-a^2))}$. I know that $\displaystyle \int_\Gamma \frac{dz}{z}=2\pi i$. Now, how do I get rid of $\sin t \cos t$ part? I am stuck here. Can somebody please explain how? What is the Physical Meaning of Commutation of Two Operators? What exactly do \csname and \endcsname do? What exactly do \csname and \endcsname do? What are their job? I have glanced at the Texbook and some other books, but none of them was clear enough to me. Can anyone please give a simple example to clarify this issue? LaTeX to plain text for e.g. generation of statistics I would like to convert a large LaTeX project (i.e. spanning multiple files) into plain text. The purpose is generation of statistics, so representing mathematics is not an issue. In fact, all mathematics is ideally ignored.  I have found  but could not get it to run. It seems unfinished, but is exactly what I am looking for otherwise. How to convert aspell dictionary to simple list of words? I want to get list of all words from aspell dictionary. I downloaded aspell and aspell polish dictionary, then unziped it using:  preunzip pl.cwl   I got pl.wl:  ... hippie hippies hippiesowski/bXxYc hippika/MNn hippis/NOqsT hippisiara/MnN hippiska/mMN hippisowski/bXxYc ...   but they appear with sufix like /bXxYc or /MNn. These suffixes are defined in pl_affix.dat, which looks like  ... SFX n Y 5 SFX n   a         0         [^ij]a SFX n   ja        yj        [^aeijoóuy]ja SFX n   a         0         [aeijoóuy]ja SFX n   ia        ij        [^drt]ia SFX n   ia        yj        [drt]ia ...   It is connected to the declination and conjugation. How can I add to the first list all forms (with all corresponding suffixes as defined in .dat file ) ?  BTW: I need this list to spell-checker jazzy. Reverse Cauchy Schwarz for integrals Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that $$\sqrt{\int_a^bf^2 \int_a^b g^2}\leq \frac{1}{2}\left(\sqrt{\frac{M_1M_2}{m_1m_2}}+\sqrt{\frac{m_1m_2}{M_1M_2}}\right)\int_a^bfg$$  I can't make any significant progress on this one... Thanks for any hint. Is it possible to use text-overflow:ellipsis on multiline text? Interpret geometrically the equation The function $f$ is convex when:  $$f(tx+ (1-t)x', ty + (1-t)y') ≤  tf(x,y) + (1-t)f(x', y')$$  for all numbers $0≤t≤1$ and all pairs of points $(x,y)$ and $(x',y')$ in the domain of $f$ (the whole of $R^2$)  How would I interpret this geometrically? Also what is an example of a convex function whose graph is not a plane? When & why I need to use web.AllowUnsafeUpdates = true How to pad zeroes to a string? Change the Fill Color of a cell if the value of that cell equals the value of another specific cell I want to change the fill color of cell C5 if it has the same value as cell A5.   Can this be done? I know how to format cell C5 if it has a certain value - but not when it is to be compared to the value of another cell. BibTeX bibliography not printing I was unable to find a previous post about this - perhaps too simple? I just came across the BibTeX and am trying to use it with an EndNote library. I have exported to the .bib file (which looks good as far as I can tell), but no bibliography is printed and the references aren't being cited. Obviously there is something major missing.  My document looks as follows.  \documentclass[11pt, fceqn]{article} \usepackage{epsfig} \usepackage{setspace, titlesec, caption, floatrow}    \usepackage{amsmath, amssymb, amsfonts, amsthm, cite, mathtools, lineno, subfig, natbib}  \begin{document}   Some text \cite{Alo_1}.  \bibliography{bib} \bibliographystyle{plain}  \end{document}   Any thoughts would be greatly appreciated. Proof of a proposition about recursion definition (Terence Tao's Analysis I) Exercise 3.5.12. Let $ f: \mathbb N \times \mathbb N \to \mathbb N$ be a function, and let c be a natural number. Show that there exists a function $a:\mathbb N\to \mathbb N$ such that $$a(0)=c$$ and $$a(n++)=f(n,a(n))\;for\;all\;n\in \mathbb N,$$ and furthermore that this function is unique. Tao gives the hint: first show inductively, by a modification of the proof of Lemma 3.5.12, that for every natural number $N\in \mathbb N$, there exists a unique function $a_N:\{ n\in \mathbb N:n\lt N\}\to \mathbb N $ such that $a_N(0)=c$ and $ a_N(n++)=f(n,a(n))$ for all $n\in \mathbb N$ such that $n\lt N$.  In my opinion, we could solve it by proving that $\forall n\in \mathbb N,a(n)\in \mathbb N$ according to mathematcial induction.  The property P(n) is:$$\forall n\in \mathbb N, a(n)\;is\;unique$$ P(0) is true. And if P(n) holds, P(n++) is true because different natural numbers must have different successors. $$a(n++)=f(n,a(n))$$ $$a(m++)=f(m,a(m))$$ $$n++=m++\Rightarrow n=m\Rightarrow a(n++)=a(m++)$$ It's similar to the proof of  Proposition 2.1.16 (Recursive definitions): Suppose for each natural number $n$, we have some function $f_{n} : \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to the natural numbers. Let $c$ be a natural number. Then we can assign a unique natural number $a_{n}$ to each natural number $n$, such that $a_{0} = c$ and $a_{n{++}} = f_{n} (a_{n})$ for each natural number $n$.  This proof indicates both the existence and uniqueness of the function $a$. I don't know why Tao says it's informal.  But regarding the hint, how do we get the function f even if we've got the conclusion from the hint. Also it seems like the function a(n) can be diffentent considering different N. I don't know if the hint is correct. Adding words to links generated by hyperref \documentclass{article} \usepackage{hyperref} \usepackage{lipsum} \usepackage{polski} \begin{document} \begin{figure} . \caption{A figure} \label{fff} \end{figure} \lipsum[1]  Rysunek~\ref{fff} jest zabawny, bo na rysunku~\ref{fff} nic nie ma. \end{document}   In this sample, a clickable link is added to output PDF for 1 and 1. I want to be able to change them to Rysunek 1 and rysunku 1. I want to be able to put any other text + number of the figure into a hyperlink. The word rysunek can't be set in preamble as a default because there is a declension in Polish. The same question goes for tables, listings, equations, pages (pageref) etc (basically any reference).  HTML equivalent of my MWE:    Rysunek &lt;href="fff"&gt;1&lt;/a&gt; jest zabawny, bo na rysunku &lt;a href="fff"&gt;1&lt;/a&gt; nic nie ma.   I want it to be:    &lt;href="fff"&gt;Rysunek 1&lt;/a&gt; jest zabawny, bo na &lt;a href="fff"&gt;rysunku 1&lt;/a&gt; nic nie ma. How to programmatically bind a (dependency) property of a control that's inside a DataTemplate? Can an Indian citizen visit Dubai on a layover? I'm trying to work out this shoulder deformation. I can't figure it out.  Is there anything wrong in my armature? Should i add bones and keep more control of my mesh or use shape keys? Here is my file blend. Area51 proposals are being advertised now all over the network? While happily browsing  I suddenly noticed this:        Curious, I clicked and landed . Appears to be ordinary Area51 proposal, one out of hundreds.  Do all proposals get such ad at some point? What is the criteria? Any way to ask for such ad?