Showing the factor ring $\mathbb Z[x]/\langle x^2 + 1\rangle$ is isomorphic to the Gaussian integers.
Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$?
Include the "People Reached" and "Helpful Flags (count)" data in SEDE I know the new profile design hasn't been updated for  on the network yet, so maybe this is a bit too early. Nonetheless, please include the &quot;People Reached&quot; data in  at your earliest convenience. Also, I was just made aware that users helpful flag counts are now publicly displayed on their  towards the top-right, underneath the people reached stat. So please include this data in SEDE as well. How do I drag and drop files into an application? I've seen this done in Borland's  environment, but I'm not sure how to go about it for a C# application I'm working on. Are there best practices or gotchas to look out for? If I have several classes with functions that I need but want to store separately for organisation, can I extend a class to have both?  i.e. class a extends b extends c  edit: I know how to extend classes one at a time, but I'm looking for a method to instantly extend a class using multiple base classes - AFAIK you can't do this in PHP but there should be ways around it without resorting to class c extends b, class b extends a How often do publishers sue researchers for copyright infringement for putting their articles on a personal website? I know that some publishers sometimes send copyright infringement emails to researchers putting their articles on a personal website. How often do publishers actually sue researchers for copyright infringement for putting their articles on a personal website? (assuming the authors didn't have the legal right to do so)  I am mostly interested in United States-based authors, but I am curious about other countries as well. Python scope: "UnboundLocalError: local variable 'c' referenced before assignment" I am trying to figure out this:  c = 1 def f(n):     print c + n  def g(n):     c = c + n  f(1) # =&gt; 2 g(1) # =&gt; UnboundLocalError: local variable 'c' referenced before assignment   Thanks! How to play MP3 files using Rhythmbox? Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics Suppose $K/F$ is an extension field, and $E_1$ and $E_2$ are two intermediate subfields of finite degree. Let $E$ be the subfields of $K$ generated by $E_1$ and $E_2$. I'm trying to prove that $$ [E:F]\leq[E_1:F][E_2:F].$$  Since $E_1$ and $E_2$ are finite extensions, I suppose they have bases $\{a_1,\dots,a_n\}$ and $\{b_1,\dots,b_m\}$, respectively. If $E_1=F$ or $E_2=F$, then the inequality is actually equality, so I suppose both are proper extension fields. I think $E=F(a_1,\dots,a_n,b_1,\dots,b_m)$. Since $\{a_1,\dots,a_n,b_1,\dots,b_m\}$ is a spanning set for $E$ over $F$,  $$[E:F]\leq n+m\leq nm=[E_1:F][E_2:F]$$ since $m,n&gt;1$.  Is this sufficient? I'm weirded out since the problem did not ask to show $[E:F]\leq [E_1:F]+[E_2:F]$ which I feel will generally be a better upper bound. Interview Question Asked In yahoo Can you find the smallest positive number such that if you shuffle the digits of the number in a particular order, the shuffled number becomes twice the original number.  Source:   I understand the answer is $125874 =&gt; 251748$  $251748$ is twice the $125874$ and have same digits $1,2,4,5,7$ &amp; $8$  but how to solve this non programmatic ? Should you pick a baby up everytime it cries? My husband and i have differing opinions on this. I don't believe in the term 'spoilt'. I would be very happy if my daughter was 'spoilt' in the sense that she expects to be treated with the highest possible respect and love and affection in all her relationships.   Also it would not bother me if she think she is special and unique (that's exactly what I want her to think). Mother's face is a mirror for the baby - through the mother's reactions and attention, the child learns that she is a lovable and valuable human being.   I spend all my time with my baby when she is awake. I play games with her, I smile, and laugh, and I never ever let her cry. I rush to pick her up immediately, even if it's the middle of the night, I force myself to get up, no matter how tired I am (I admit it's hard sometimes, and I do let her cry for 10-20 seconds sometimes, but never more). Detecting and acting on keyboard direction keys in Java To be truly standards-compliant, must all functions in C (except for main) have a prototype, even if they are only used after their definition in the same translation unit? Evaluate $\prod \frac {2k - 1} {2k}$ I came across to this: evaluate $$\prod_{k = 1}^{n} \frac {2k - 1} {2k}.$$  I brought it to a closed-form expression in asymptotics but I suspect my asymptotics is bad and I also want to see different solutions, hence the question.  My original solution: Write   $$\begin {eqnarray} \prod_{k = 1}^{n} \frac {2k - 1} {2k} &amp; = &amp; \exp \left (\sum_{k = 1}^{n} \log \left (1 - \frac {1} {2k}\right) \right) \nonumber \\ &amp; = &amp; \exp \left (- \sum_{k = 1}^{n} \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m}\right). \end {eqnarray}$$  Let $$S = \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m} \tag {1}.$$ Multiply both sides of $(1)$ by $2k$ to get $2k S &lt; S + 1$, and by $4k$ to get $4kS &gt; S + 2$. Hence, $$\frac {2} {4k - 1} &lt; S &lt; \frac {1} {2k - 1}.$$ Then, $$\frac {1} {2} \left( \log (4n - 1) - \log 3 \right) - 2 + O \left(\frac {1} {n^2}\right) &lt; \sum_{k = 1}^{n} \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m} &lt; \frac {1} {2} \log (2n - 1) - 1 + O \left(\frac {1} {n^2}\right).$$ Hence, $$\sqrt {\frac {c} {2n - 1}} &lt; \prod_{k = 1}^{n} \frac {2k - 1} {2k} &lt; \sqrt {\frac {3 c^2} {4n - 1}},$$ where $c$ is a constant. Any better approach? Prove $2^k+1$ divisible by 3 for odd K Mounting a device — role of /dev, /media and /mnt, and the mount command I have several closely related questions about what happens when I insert a CD. The files on the CD /media/Ubuntu\ 11.04\ i386/, but from what I've seen /dev/cdrom is also involved.   What is the difference between /dev, /media and /mnt? Following is what I have found from internet but I still have little idea:        /dev — this folder contains device   files    /media — this is a mount point for   removable devices   /mnt — this is a temporary mount point     What is the purpose of mount? In other words, if a device has been represented by the OS as a device file under /dev, why can it not be accessed via the device file directly without mounting?   Is mount only used for storage device, not for non-storage device, such as graphical card, network card, camera, ...? Where is a device file under /dev mounted to, under /media or under /mnt? I remember I have seen both, but am curious when to mount to which? I found my CD was automatically mounted to /media/Ubuntu 11.04 i386. I guess the device file of the CD is /dev/cdrom, but I cannot confirm it by looking into /dev/cdrom and /media/Ubuntu 11.04 i386:  $ ls -l /media/Ubuntu\ 11.04\ i386/ total 3522 -r--r--r-- 1 Tim Tim     143 2011-04-27 13:04 autorun.inf ... $ ls -l /dev/cdrw lrwxrwxrwx 1 root root 3 2011-05-28 15:12 /dev/cdrw -&gt; sr0 $ ls -l /dev/cdrom lrwxrwxrwx 1 root root 3 2011-05-28 15:12 /dev/cdrom -&gt; sr0   How can I find out which device file is for my CD? This started to happen immediately after I had rebooted the first time after doing a system upgrade. It first starts with a dialogue that says "System program problem detected". Then when I try to hit 'report problem' not much happens. I am led through a dialogue that always ends up the problem cannot be solved.  I am aware this is not a lot of information, however I'm not sure which information I need to publish and how should I obtain it to debug this problem.  Here's a screenshot! Dropping one of the columns when using one-hot encoding My understanding is that in machine learning it can be a problem if your dataset has highly correlated features, as they effectively encode the same information.  Recently someone pointed out that when you do one-hot encoding on a categorical variable you end up with correlated features, so you should drop one of them as a "reference".  For example, encoding gender as two variables, is_male and is_female, produces two features which are perfectly negatively correlated, so they suggested just using one of them, effectively setting the baseline to say male, and then seeing if the is_female column is important in the predictive algorithm.  That made sense to me but I haven't found anything online to suggest this may be the case, so is this wrong or am I missing something?  Possible (unanswered) duplicate: Show that $\frac{(2n)!!}{(2n+1)!!}$ converges. Tensor product vs. Cartesian Product for composite quantum systems machine wakes only on local magic packet I've got a Debian server, which I would like to wake up remotely using a tool like wakeonlan.  I have assigned the server a static internal IP 192.168.0.14 and set up the appropriate port forwarding in the router (which otherwise has a very limited admin interface).  When I am logged in to the same network, I can wake the server using the subnet IP, but I cannot wake the server using the public IP of the router.  What I do not understand: When the server is running and I listen on the server with netcat nc -ulv -p 9, I can see the packet coming in in both cases (both when sending to the subnet IP and when sending to the public IP of the router).  If the packet arrives in both cases, why does the machine wake up only in one of them?