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Adding Arabic text to an English document Isomorphisms preserving integral domains and fields Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following:  a). If $R$ is an integral domain then $S$ is an integral domain.  For this, I said that If I let $f: R \to S$ be an isomorphism and $(R, + , \circ), (S, \oplus, \ast)$, then let $x, y\in R$,  such that $x \circ y = 0_R$, because $f$ is an isomorphism, $f(x \circ y) = f(x) \ast f(y) = f(0_R)$. If I let $R$ be an integral domain then, as per the definition $R$ is commutative, has unity, and no zero divisors.   So $f(x \circ y) = f(x) \ast f(y) = f(0_R)$ where we have either $f(x \circ y) = f(0_R) \ast f(y) = f(0_R)$ or $f(x \circ y) = f(x) \ast f(0_R) = f(0_R)$.   Since $f$ is an isomomorphism, let $s = f(x), t = f(y)| s,t \in S$. Then $s \ast t = f(0_R)$. By definition, $S$ also has unity so $0_S$ exists. If we map $0_R \to 0_S$. Then either $s = 0_S$ or $t = 0_S$.   So then, since $S$ has unity, is commutative, and has a no zero divisors as we showed above, it is also an Integral domain.  I'm not sure I have this correctly.  b). If $R$ is a field then $S$ is a field. I don't really understand how to do this one. Since $S$ is an integral domain, it has a $0_S$. So I need to show that $S$ distributes $\ast$ over $\oplus$.  Any help with these is appreciated Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Is there a license / royalty free DRM available for protecting and distributing video content? Solve this integral for free WiFi The sum of an uncountable number of positive numbers Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha&lt;\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).  Proof: Let $\sum_{\alpha\in A}x_\alpha=M&lt;\infty$. Consider $S_n=\{\alpha\in A \mid x_\alpha&gt;1/n\}$.  Then $M\geq\sum_{\alpha\in S_n}x_\alpha&gt;\sum_{\alpha\in S_n}1/n=\frac{N}{n}$, where $N\in\mathbb{N}\cup\{\infty\}$ is the number of elements in $S_n$.  Thus $S_n$ has at most $Mn$ elements.   Hence $\{\alpha\in A \mid x_\alpha&gt;0\}=\bigcup_{n\in\mathbb{N}}S_n$ is countable as the countable union of finite sets. $\square$  First, is my proof correct? Second, are there more concise/elegant proofs? The internal storage on my phone is almost full. The storage screen tells me    2.77GB / 2.91 GB taken Apps 0.89GB Other 1.89 GB Available 138MB   What is this "other" and how can I clean it up? I already followed the steps of "make more space", it barely makes a difference.   Phone Storage then tells me 43.00MB / 1.09GB is taken.  And on the SD card I have 1.86GB / 14.72GB taken.  I have plenty of storage in various places yet when I try to install apps I'm unable to do so cause of "insufficient storage". How can I fix this? How do I recover my Google account (or Gmail) password or username? I have forgotten the password or username for my Google account, or I'm unable to access my account for other reasons.  How can I recover my account?  (This Q&amp;A is meant as a general description for all Google and Gmail username/password questions. See: ) "no public key available" on apt-get update I got to read Volume I of Feynmann's lectures. It said that at absolute zero, molecular motion doesn't cease at all, because if that happens, we will be able to make precise determination of position and momentum of the atom. We do know that Heisenberg uncertainty principle holds for microscopic particles in motion. But what then is wrong to consider that all molecular motion ceases at absolute zero? In other words, does the uncertainty principle not hold when there is no motion?  Need some help! continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$ Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$.   $g(0)=1$. If $a=g(1)$,then $a&gt;0$ and $g(r)&gt;a^r$ for all $r$ in $\mathbb Q$.   Show that the function is strictly increasing if $g(1)$ is greater than $1$, constant if $g(1)$ is equal to $1$ or strictly decreasing if $g(1)$ is between zero and one, when $g$ is continuous. Let us say you are in an inertial reference frame with a circular planar disk. If you take your meter measuring rods (or perhaps tape measure) you can find the diameter and circumference of the disk. If you divide the circumference by the diameter, you will get exactly $\pi$. Now you start rotating the disk. A book I have claims that now the ratio will be different (lets call it $\pi_\circ$ to avoid ambiguity.) This has caused to reevaluate my understanding of special relativity (this is a set up to general relativity, but the problem itself only requires the special theory.) My problem is that, sure, the measuring rods (or tape) would contract on the circumference, but wouldn't the circumference contract a similar amount, therefore giving a measurement of $\pi$ in every reference frame according to your own measuring rods. Of course, this reasoning goes on to make me question how I understood all of special relativity up this point. Suffixes indicating people and language of country vary by country:  _an: American, German, Italian, Belgian, Australian, Russian, Ukrainian, Korean,  Mexican, Brazilian, Chilean, Argentinean, Indonesian, Malaysian, Iranian, Egyptian, Somalian, Palestinian  _sh: English, British, Polish, Danish, Swedish, Turkish,    _ch: French, Dutch  _ese: Chinese, Taiwanese, Japanese, Vietnamese, Portuguese  _no: Filipino, Latino  _i: Israeli  _: Thai  Among the above, _an seems to outnumber other suffixes.  Are there any simple rule to govern appropriation of suffix turning a country name into adjective, indicating people and language of countries in the world in terms of say, region, race, or cultural background? Or is it just discretional? Add title uniqueness to the heuristics for detecting low-quality questions This feature request is a response to  Two days ago,  suggested that we . However, given that  (hat tip ) as well as ...  262  103   ... I suggest that we take it a step further. Let's extend the heuristics for detecting—and blocking—low-quality questions to enforce unique titles, or at least flag new questions with duplicate titles automatically. We can leave the precise heuristics of detecting a duplicate title up to the Stack Overflow development team. I have no delusions about the fact that a few questions will &quot;slip through the cracks.&quot; The goal here is to be proactive about minimizing the number of duplicates, and I think that the best place to start is the existing heuristics for detecting low-quality questions. After all, duplicate question titles are bad for a number of reasons: They often indicate a question that is itself a duplicate, but perhaps just as importantly, they decrease the signal-to-noise ratio when searching for questions. I honestly can't think of any reason that duplicate question titles are helpful, unless you are asking for a solution to the same problem in a different programming language. Even then, I think that it would be acceptable to rephrase the question to say &quot;How do I do X in (insert programming language here)?&quot;  makes a good point about the potential problems with this solution:  I have a hunch that if we do that, people will find ways to circumvent it (changing case or punctuation, adding random spam), and we'll end up stuck in a an arms race.  We may get stuck in an arms race with some users, but I honestly think that many people new to Stack Overflow just don't understand the etiquette of asking questions here. If we could even convince people to rephrase their question more clearly 20% of the time, I think that it would be a tremendous boon to the community. Furthermore, remember that heuristics for low-quality questions already exist, arms race or no. If we look at light as a collection of little particles, we can say that dimmer light has its photons more spread out.  But if we look at light as a wave, then there no gaps unless specifically placed there on purpose.  So how can one interpret the formation of gaps between the photons? What if the universe is rotating as a whole? How to connect a GUI button to function? "Taste" is to "flavor" as "touch" and "sight" are to what? For the senses, we have:    flavor for taste aroma/odor/scent for smell sound for hearing ____? for touch/feel ____? for sight/see   So one tastes a flavor, smells an aroma, hears a sound, feels a(n) _____, and sees a(n) _____. For the former, part of me wants to say texture, but I feel that is too specific; for the latter, I want to use visual or sight, but but does that make sense, seeing a visual or a sight? I have recently upgraded SQL Server 2000 database to 2008 R2.    What I did was:   Shutdown SQL Server 2000 (express) service on old machine, Move datafiles (mydatabase.mdf and mydatabase.ldf) to new machine, Run SQL Server Management Studio 2008, Connect to local database engine, Attach datafiles to database. Change the compatibility level of database to SQL 2008 (100).     Question: What else should I do to make migration complete?  I want to:   use new features like checksumming and full recovery model, make this database to be exactly as it was created in SQL 2008 R2, make this database to be fully compatibile, correct and be perfect suited for new, SQL 2008 R2 database engine.     In other words: I just want to know how to correctly and completly convert old SQL 2000 database to new 2008 R2 database, be calm that everything is done right and be happy with all new features.    I'm asking this question, because I've found a lot of sites on the Internet that says so many different things that makes me confused: some say that it is required to rebuild indexes, another says to do other things... and now I know nothing so I want to hear experienced person opinion and clear, step-by-step instructions. I work for very small company, I'm on my own and I don't want to screw things up.    Sir, I'm really impressed with your answer, I wasn't expecting so much.    So some comments:   The database is now in production. As I said, it was upgraded using deattach-attach method as I desribed in first post and as described on MSDN:  It had to be done quickly, so I was forced to do it that way. Let's forget about how inappropriate could it was and focus on current situation. The users/persmission is not a problem here - there are only few and permissions are simple. The application which use database is compatibile with SQL 2000 till 2012 so this is not a problem either. The database file (MDF) isn't big - only about 1GB.     Few more questions:   You recommend to use backup/restore method, but I did as written above, so can I encounter any problems now? Everything worked without any problem. About checksumming and full recovery model: it wasn't available/enabled on SQL 2000, so I want to use them now. You said that the only thing I need to do is to enable those options in database properties? I've read somewhere, that it isn't enough and I should also rebuild indexes or something. I really don't know, I just asking. I'm prepairing to migrate this database to SQL 2012 - so first it was from SQL 2000 to 2008 R2, now it will be from 2008 R2 to 2012 (it was impossible to do this directly because of lack of support of SQL 2000 databases in SQL 2012). So I understand that I should follow your guide: backup it in 2008 R2 and restore in 2012, then do the rest of your tips, right? Please explain me the backup/restore method: Is it like a dump of database to SQL queries and then restoring it by executing a bunch of queries? Will this method by the way "defragment" my database? If not, how to defragment/optimize it manually? As we were using SQL 2000 Express for years (no management interface), we were doing backups simply by stopping engine and RAR the DATA directory. For now, as we are on SQL 2008, isn't this still better than using backup function in Management Studio? Full recovery mode with frequent Transaction log backups - Where is the Transaction log stored - is it the LDF file? How sould I backup it properly?     I know that my questions may sound silly, I'm not professional database admin, but I am the only person here who can do such "hard core" task like upgrading the database engine. I'm also sure that your knowleadge will help a lot to other people like me.  Thank you very much for your time and knowleadge, I really appreciate this. For the command   sudo apt-get update   I get following error:  W: Failed to fetch http://ch.archive.ubuntu.com/ubuntu/dists/maverick-updates/multiverse/binary-i386/Packages.gz   407  Proxy Authentication Required (The ISA Server requires authorization to fulfill the request. Access to the Web Proxy filter is denied. )   I am running Ubuntu 10.10 installed on Windows XP using VirtualBox. For Internet connections I am using proxy server with an authentication. I tried to use gnome-network-proxy tool to set proxy settings system-wide.   After that /etc/environment has been updated by http_proxy variable with the format http://my_proxy:port/, there were no authentication data.  I checked this with Firefox. Browser asked my for login and password and everything was working fine. It was unfortunately not the case for apt-get. I have also tried to do as described . Unfortunately it does not work.  May it be somehow related to the fact that a proxy is in a Windows domain, any ideas?  My proxy name is http-proxy. Is '-' a special character here?