Adding some more rigour to a simple exercise in Hartshorne I have just begun working through Hartshorne's algebraic geometry.   In part (b) of exercise 1.1, we need to show that $A(Z)$ is not isomorphic to a polynomial ring, where $Z$ is the plane curve $xy =1$.   Since $k[x,y]/\langle xy - 1 \rangle\cong k[x, \frac{1}{x}]$, it makes sense that $A(Z)$ is not isomorphic to a polynomial ring. But how do  I rigorously prove that $$k[x, \frac{1}{x}] \not \cong k[x,x^n]$$ for some $n$?
Show that quotient rings are not isomorphic I've been given a homework problem that requires me to show that the rings $\mathbb{C}[x,y]/(y - x^2)$ and $\mathbb{C}[x,y]/(xy-1)$ are not isomorphic.    This is my attempt at a solution:  For $\mathbb{C}[x,y]/(y - x^2)$, we can parametrize in the following way: $x = t$ and $y = t^2$. Then this ring is isomorphic to $\mathbb{C}[t]$.  For $\mathbb{C}[x,y]/(xy-1)$, we can parametrize $x = t$ and $ y = 1/t$. Then this is isomorphic to $\mathbb{C}[t, 1/t]$.   But $\mathbb{C}[t, 1/t]$ is not isomorphic to $\mathbb{C}[t]$.  Am I on the right track? If not, any helpful hints?
Can I travel domestically within the United States with my USA expired passport? I need to fly domestically within the United States. The only ID I have is an expired USA passport.  Can I use that to pass T.S.A.?  Can I use it as ID? Why is AES resistant to known-plaintext attacks? I'm having trouble solving a Fitch Style Proof and I was hoping someone would be able to help me.  Premises: p ∧ q  Goal: q ∨ r  Would be an assumption (p ∧ q => q ∨ r) in first step a correct one?   Then, q ∨ r Implication Elimination: 2, 1  Why is it not correct?  Thank you very much for any help! How do I rename a directory via the command line? I have got the directory /home/user/oldname and I want to rename it to /home/user/newname. How can I do this in a terminal? Why do you lose reputation for downvoting answers? Pairwise independence of Random variables does not imply indendence Show by a counterexample that for a family $(X_i)_{i\in I}$ of random variables the independence of all pairs $(X_i,X_j)$ with $i,j\in I, i\neq j$ does not imply the independence of the family (It is enough to consider $\mbox{card}(I)=3$.).      Hello, I am searching for a simple example. The professor gave the hint that one might consider random variables which only can take two different values, but I do not know exactly what is meant.  By the way:  We call the family $(X_i)_{i\in I}$ of random variables independent, if $(\sigma(X_i))_{i\in I}$ is independent, whereat we call a family $(\mathcal{E}_i)_{i\in I}$ with $\mathcal{E}_i\subset\mathcal{A}$ independent, if for any finite subset $I_0\subset I$ and any choice $E_i\in\mathcal{E}_i, i\in I_0$, it is $$ \mathbb{P}(\bigcap_{i\in I_0}E_i)=\prod_{i\in I_0}\mathbb{P}(E_i). $$ Hello,    This question is about the Command Line Developer Tools that are usually installed with xcode-select --install and updated via a software update from the Mac App Store (at least until macOS 10.13). I use the developer toolchain on a daily basis and it has always worked and updated without issues.    Today I've updated my mac from High Sierra 10.13.6 to Mojave 10.14.1, and I've lost the ability to update the Developer Tools.    After the update I've executed a terminal command that relies on the developer tools being installed. It was a command to update , although I don't think that the specifics matter, as I believe that any task that tried to access the developer tools would have triggered the same error message.    The error was:   xcrun: error: invalid active developer path (/Library/Developer/CommandLineTools), missing xcrun at: /Library/Developer/CommandLineTools/usr/bin/xcrun       The phrasing was new (maybe), but I thought that it was caused by the usual need to re-install the Developer Tools after some macOS updates.    Later I tried to dig a bit deeper into the failure, and found out that:  $ ls -l /Library/Developer/CommandLineTools/usr/total 0drwxr-xr-x  3 root  admin  96  4 Nov 19:32 share$ which xcrun/usr/bin/xcrun$ xcode-select -p/Library/Developer/CommandLineTools    Still, as I normally do, I entered the command to install and update the Developer Tools in the terminal:  xcode-select --install    This started the usual procedure: modal window to confirm, then request to accept the license agreement, followed by a progress bar. Except that it failed very quickly with an unexpected error:     Can&rsquo;t install the software because it is not currently available from the Software Update server.     I've tried several times, to no avail. It gets always stuck.    Sometimes, however, the failure causes the System Preferences to report a pending update:      If I open that preference panel it starts searching for updates, and it always finds nothing except the first time that it happened. The first time, it found this:          It literally suggested me to install the updates for macOS 10.11 and 10.13. I have no idea why.  I closed the preference without installing, and as I said it hasn't shown them again. There is also no update available in new Mac App Store.   Is this a known issue? Is there any way to resolve the problem?     Update  Of course I can download the Dev Tools for macOS 10.14 installer from .  Before installing them manually, though, I'm wondering if there is anything that's broken with the system. Geometric interpretation of Electromagnetism For gravity, we have    General Relativity, which is a geometric theory for gravitation.  Is there a similar analog for Electromagnetism? Any command that starts with $ returns  command not found   How can I fix this?  I am  using 16.10 Ubuntu. I am trying to create a beamer animation using tikz, but I can't even get basic rotating of a tikz image working as desired.  I have a rectangle specified by  \draw[rounded corners,dotted] (-3.5,0) rectangle (-0.5,2);   I would like to rotate this rectangle about the point (-1,.5) (which is near the bottom right corner of the rectangle).  I've tried:  {\rotatebox[origin=c,x=-1,y=.5]{130}{             \draw[rounded corners,dotted] (-3.5,0) rectangle (-0.5,2);}}   But this produces an error.  If I just use the origin=c option, it compiles, but it rotates the rectangle about the point (0,0).  I've also tried it with just the x=-1,y=.5 (without the origin option), and it fails to compile.  I have searched the documentation, which is where I learned about the options x=xdim and y=ydim.  However, I can't find a single example demonstrating how these options should be used.  Any suggestions would be greatly appreciated.    Also, if there is a better command for rotating things that will be more suitable for use in a beamer animation, please let me know.  Thanks! Where can I ask questions about the computer problems I have been facing?  Current problems I am facing with my computers:   Windows 7 errors random "Blue screens of Death" computer randomly freezing up Xfire challenges Arriving at a different port of entry in the same state in USA? How can I classify data with the nearest-neighbor algorithm using Python? How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ How   $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$   i have tried to do that by the following procedure:  $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - n\bar X^2)$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - \sum_{i=1}^n\bar X^2)$  =$\frac{1}{n} \sum_{i=1}^n (X_i^2 - \bar X^2)$  Then i have stumbled. Inequality $\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$ If $x&gt;0$, $y&gt;0$, $z&gt;0$ and $xyz = 1$ then   $$\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$$   I tried using $\displaystyle x = \frac{a}{b},y = \frac{b}{c}$ and $\displaystyle z = \frac{c}{a}$ substitution,  $\displaystyle \sum_{cyc} \frac{\sqrt{a^2+b^2}}{b} \le \sqrt{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right) \\ \displaystyle  \iff 6+ 2\sum_{cyc}\frac{\sqrt{(a^2+b^2)(b^2+c^2)}}{bc} \le \sum_{cyc}\frac{a^2+b^2}{b^2} + 4\sum_{cyc}\frac{b}{a}$  So it would suffice if we showed $\displaystyle \sum_{cyc}\frac{\sqrt{(a^2+b^2)(b^2+c^2)}}{bc} \le 2\sum_{cyc}\frac{b}{a}$  Squaring again does not lead anywhere nice. Is the last inequality true ? If not is there a better way of showing the result ? Are these plural or singular? I was surprised when I heard zero is plural, and even 1.00001 is plural.  Then, what about following numbers ?  (1) 1.00  which means the value measured  between 0.995-1.004, having possibility that it was exactly 1.  (2) 0.99999.....  which is mathematically equal to 1. Also, are others that are mathemattically equal to 1 than this, treated completely same as 1?  (3) -1  Is this singular? Or negative numbers are generally plural?  (4) i = square root of -1  I know it's weird to count things with imaginary numbers, nonetheless, if you were to count, which form, plural or singular, would you be to use?  That's it. Create bootable CD from ISO Decode HTML entities in Python string? Coordinate descent soft-thresholding update operator for LASSO I was reading this paper () describing the coordinate descent algorithm for LASSO, and I can't quite figure out how the soft-thresholding update for each $\beta_j$ is derived for the linear regression case. Specifically, how does one go from equation (4) to equation (5) in the paper? Can anyone show me a detailed derivation of the update? Thanks!  Edit: actually this paper deals with the elastic net penalizing term, which reduces to LASSO when $\alpha=1$. I am mostly interested in LASSO, but I guess the derivation should be principally the same for elastic net. Render turns out plain white For some reason, when I render out the image, it is plain white. I've checked the sky settings, and I tried applying textures, but nothing seems to change.  How can I fix this?