What set is $\mathbb{Q}(\sqrt{2},\sqrt{3})$? Is it the set $X = \{a\sqrt{2}+b\sqrt{3}:a,b\in\mathbb{Q}\}$?
I know that : $\mathbf{Q}(\sqrt{2}) = \mathbf{Q}+ \sqrt{2} \mathbf{Q}$ , but then what is $\mathbf{Q}(\sqrt{2},\sqrt{3})$?
How to enter the Default Keyring password via the command line? Is there a way to enter the default keyring password using the command line?   For instance:  You have a remote setup of Ubuntu 10.10 thats set to auto login. You don't want to remove  the keyring password.  All right the system boots up and logs in automatically, then asks for the keyring password now at this point you can create ssh connections but you can't remote desktop.   What can you do to enter the keyring password at this point?  Also, to better clarify, this is from a remote connection using the command line. No, actually I didn't, but that was a close call and I'm really curious how would one go about that one.  On my third or fourth visit to the Nether, I still didn't realize the risk, and didn't place marks that would help me get back to the origin (currently I like to use flint and iron to create big flaming arrows towards the exit, visible from afar.) I wandered past one hill, and then I couldn't locate the place I came from. I wisened up and started placing marks in form of torches, towers of netherrack with glowstones on top etc. The bunch of wood I had in my stash has proven invaluable as I ran out of steaks, and had to craft a bowl and start making mushroom soup. I realized the fastest way back would be to simply die, but since I had no clue where I was, that would mean losing my quite precious equipment, which cost me a lot of time and effort to gather and enchant, for that trip specifically. I also knew I had traveled rather far from the portal before starting placing marks, so the chances of finding it again by going in any random direction were rather slim.  By sheer luck I found one of the caverns I had explored before, and found my way back, but that was really a close call and I'd prefer to know any hints what to do in Nether if you're already lost (no Obsidian to build a new portal, not willing to die and forgot the direction to the exit.) C programming, why does this large array declaration produce a segmentation fault? Some icons are correct, and some are whack. Turns out it's not just the hot questions list, but the whole icon set. How to take the gradient of the quadratic form? It's stated that the gradient of:  $$\frac{1}{2}x^TAx - b^Tx +c$$  is   $$\frac{1}{2}A^Tx + \frac{1}{2}Ax - b$$  How do you grind out this equation? Or specifically, how do you get from $x^TAx$ to $A^Tx + Ax$? I am planning to visit the UK on a USA travel document but there’s nowhere I am able to find information about visas: I’m a Zimbabwean but using a USA refugee travel document do I need to apply for a UK visa before I travel or I can they just give them on arrival. Is there a detailed user's documentation for KDE Plasma 5? Compute the moment generating function of $Y = X_1X_2 + X_1X_3 + X_2X_3$ CPU time out limit was being hit while processing child records on the inline opportunity visualforce page, to handle this i wrote a remoteActionFunction on the page that invokes controller method that's taking long to process and when i get CPU LIMIT exception in the result, i am invoking javascript call back function that invokes a action function which handles the same code asynchronously(@future).  Question: by setting the timeout (Visualforce.remoting.timeout = 120000;) can't we overcome CPU TIME out limit of 10,000 ms. 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 ------------------------- Is it possible to use multiple desktop environments on same system? I am using Ubuntu 12.04. I want to try out KDE and other desktop environments.   But I'm not sure if I'll like them enough to switch permanently. I want to be safe and make a transition in a piecemeal fashion.  Is it possible to install KDE on Ubuntu and yet not do away with the default Unity shell?  Could I install multiple desktop environments and switch between them by tweaking some startup configuration and then rebooting? Find the equation of a plane containing a point and passing through a line Find the general form equation of a plane that contains the point $(2, -2, 1)$ and passes through a line $(x, y, z) = (1, 2, -3) + t(2, -3, 2)$.  I managed to use the fact that it contains a given point to write the equation of the plane as $A(x-2) + B(y+2) + C(z-1) = 0$ but got stuck there. Drop shadow for text in PGF/Beamer I want to really put an emphasis on some central piece of text I'm using in a Beamer slide.  I'm using tikz and overlay to put it, enlarged, on the center of the page, using the following code:  \begin{tikzpicture}[remember picture,overlay]   \node[at=(current page.center)] {     \scalebox{2}{\Huge\texttt{{This is important!}}}   }; \end{tikzpicture}   Can I add a drop shadow to the letter themselves, so they would look even more emphasized?  Is there some other decoration that can be added to the text itself, like a glow, etc.?  Or does PGF ignores node text altogether?  The text should be in typewriter style, so using special fancy fonts is probably not the solution. This query works for me when I tried in PG admin query tool:    UPDATE bankdetails SET bank_details = 'mysore' ,"PAN"= 123,bank_acc = 456,                        "UAN"=789,tax = 'myy' WHERE  user_id = 79; INSERT INTO bankdetails (user_id,bank_details,"PAN",bank_acc,"UAN",tax)        SELECT 79,'mysore',123,456,789,'myy'        WHERE NOT EXISTS (SELECT 1 FROM bankdetails WHERE user_id=79);   Later I implemented in my code like this:  db.query('UPDATE bankdetails SET bank_details = $2 ,"PAN"= $3,bank_acc = $4,"UAN"=$5,tax = $6 WHERE user_id = $1; INSERT INTO bankdetails (user_id,bank_details,"PAN",bank_acc,"UAN",tax) SELECT $1,$2,$3,$4,$5,$6 WHERE NOT EXISTS (SELECT 1 FROM bankdetails WHERE user_id = $1);', [companyuser.rows[0]._id, data.bankDetails.bank_details, data.bankDetails.PAN, data.bankDetails.bank_acc, data.bankDetails.UAN, data.bankDetails.tax],function(err,bank) {                  if (err) return callback(new Error('error')); })   it says:     cannot insert multiple commands into a prepared statement   Can some one guide me here? How does one attack a two-time pad (i.e. one time pad with key reuse)? My question might appear the same as the question , but actually I did read all the answers and none of them helped me with the details I need.  I am new to cryptography and my problem is with two time pad attacks on OTP. The problem I had in my course was that I have 10 ciphertexts encrypted with the same key $K$. I am then given another ciphertext that I should decrypt.  I know that XOR-ing two ciphers gives me the XOR of their original messages.  My question is what is the correct thing to do after that?  I tried to take 3 ciphertexts $C_1, C_2$ and $C_3$.  Then get $S_1 = C_1 \oplus C_2  \oplus $' ', also get $S_2 = C_1 \oplus C_3 \oplus$ ' '.  After that I compared all corresponding characters in $S_1$ and $S_2$,  and if $S_1[i] = S_2[i]$ then I calculate $S_1[i] \oplus C_2[i]$ to get $K[i]$.  I tried this on paper before coding and it worked, but I might be missing something.  Is this the right approach? Why does it work? If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example. Schwarz's Lemma and functions with positive real part Problem:     Let $D = \{ z \in \mathbb C : |z| &lt; 1 \}$. Suppose $f : D \to \mathbb C$ is a non-constant analytic function such that $\mathrm{Re} f(z) \geq 0$ for all $z \in D$.      (a) Show that $\mathrm{Re} f(z) \neq 0$ for all $z \in D$.      (b) Using a Möbius transformation and Schwarz's lemma, prove that if   $f(0) = 1$, then $$ |f(z)| \leq {1+|z| \over 1-|z|}. $$   What can be said if $f(0) \neq 1$?      (c) Show that if $f(0) = 1$, then $f$ also satisfies $$ |f(z)| \geq {1-|z| \over 1+|z|}. $$   Progress:  (a) We can use the open mapping theorem.  (b) We can compose a Möbius transformation $m$ that sends the right-half plane to $D$ with $f$. Then $h = m \circ f$ satisfies the hypotheses of Schwarz's lemma by construction. Simple manipulations of the inequality from Schwarz's lemma give the result.  (c) Looking at part (a), we know that $\mathrm{Re} f(z) \neq 0$, and so $f(z) \neq 0$. Thus, we can use the function $g = 1/f$ in the statement of part (b).  Question:  What about when $f(0) \neq 1$? I don't think you can get anything from $f(z)-f(0)+1$ unless $\mathrm{Re} f(0) &lt; 1$. And you can't in general rearrange the inputs of $f$ because $1$ isn't necessarily in the image of $f$. Also, I don't think you can get anything from $f(z)/f(0)$ unless $f(0)$ is real, because otherwise the range of $f(z)/f(0)$ could extend into the left-half plane. Why is information indestructible? In , it is established that Thieves' Cant is more of an encoding system built on top of a language than a language in its own right. According to the Basic Rules,  is:  ...a secret mix of dialect, jargon and code allows you to hide messages in seemingly normal conversation. Only another creature that knows thieves' cant understands such messages. It takes four times longer to convey such a message than it does to speak the same idea plainly. In addition, you understand a set of secret signs and symbols used to convey short, simple messages, such as whether an area is dangerous or the territory of a thieves' guild, whether loot is nearby, or whether the people in an area are easy marks or will provide a safe house for thieves on the run. (PBR, p. 27)  Can two creatures communicate using Thieves' Cant if they do not have a basic spoken language in common, or if they are speaking different basic languages? E.g. if one creature knows Orcish, Goblin, and Thieves' Cant, and another creature knows Elvish, Giant, Common, and Thieves' Cant, can they communicate with each other by embedding Thieves' Cant into their own languages? Hypothetically speaking, I can see how this could be possible in our own world, if we assume that Thieves' Cant depends on nuances of body language, emphasis, and tempo, e.g., one might say to a non-Spanish speaker:  Tenemos quEEEEEEE ir (wink) a la tieeeeenda de (shuffle left foot) armas para (twirl hair and giggle) comprar una espada (wink twice) para mi herMANO (wag right pinky and nod).  The literal message itself (&quot;We need to go to the weapon store to buy a sword for my brother.&quot;) would just be a carrier - the real message might be hidden within the gestures and the accented and extended syllables. Perhaps the drawn-out &quot;E&quot; vowel means &quot;robbery planned&quot;, the wink means &quot;bank&quot;, shuffling the left foot means &quot;be there at sunset&quot;, etc., and these are things that you wouldn't necessarily need to know Spanish to extract. Is this how Thieves' Cant actually works, or does it require actual comprehension of the literal message transmitted before the &quot;secret&quot; message can be extracted from it? Suppose the following dialogue:  ... &quot;I accept only one notion of land property. Namely, 'I am doing my stuff here, therefore I am here&quot;. &quot;But this means,&quot; he responded, &quot;you can break into any place and stay there, using this rule.&quot; &quot;There is another rule, though. Right to act in its most natural sense, 'I am here, therefore I am doing my stuff here.&quot; &quot;But do not these rules put together become circular reasoning?&quot; &quot;Of course, they do,&quot; I put my head closer to him, &quot;and this is exactly that rare case when circular reasoning is not only sufficient, but is necessary.&quot; &quot;But why is that? Circular reasoning never was a correct argument.&quot; &quot;There are many circular things happening in nature. Life forms, for example, exist only in order to produce new life forms or to prolongate their own lifetime. Is there any reason to say this can't be the case for reason itself? Why must reasoning not be circular?&quot; ...  Can circular reasoning ever be justified? For example, we know all the words in our languages can't be defined using other words without the use of circularity. We know logic can't prove itself without circularity. Are these arguments rational? Are there any other arguments where circular reasoning is still rational?