Let A = $Z$[$\sqrt{10}$] = {a+b$\sqrt{10}$$\mid$a,b $\in$ $Z$}.   Prove that any element in A an be written as a product of irreducible, but A is not a UFD.   Also a small question, I've proved that $4$+$\sqrt{10}$ and $4$-$\sqrt{10}$ are irreducible, but how to show that they cannot be associates?  Thank you so much!
I am wondering how to show that $\mathbb{Z}[\sqrt {10}]$ is not a UFD.    My only idea is to show that there are two factorizations of $10$, say, $ab, uv$ such that $a$ is not a unit times $u$ or $v$. In this ring $10=2\cdot5=\sqrt {10}\cdot \sqrt {10}$, so it suffices to show $2$ is not a unit times $\sqrt {10}$. Suppose $2=\sqrt {10}(a+b\sqrt{10})=a\sqrt{10}+10b$. Then $a=0$ since $\sqrt{10}$ is not rational. So $10b=2$, which has no integer solutions. So $\mathbb{Z}[\sqrt {10}]$ is not a UFD.  Is my reasoning correct? What are the flaws?
Complete local ring which is not complete valuation ring Complete valuation ring is obviously complete local, because valuation ring is always local ring. I would like to know the example of  Complete local ring which is not complete valuation ring  Any examples? I could find example of local ring which is not valuation ring, but the example ring is not complete. Thank you for your help. Is is possible to restore the MFT of a NTFS partition after a format? I have this problem: I'm used to compile and run .c files with CodeBlocks. If I do that from my Desktop (ext4 partition) it works without problems, but doing this from my secondary hard drive (NTFS partition) makes CodeBlocks to say permission denied while running a .c file.  I have read it's because of the partition type is it correct? But I don't know how to solve it. I tried to edit fstab (sudo gedit /etc/fstab) adding this line:   /dev/sda /media/Dati ntfs user,exec 0 1   but after rebooting Ubuntu I had a warning on boot about a problem mounting a drive, that's because I set KDE to auto mount my Dati drive on boot.   So I should auto mount my NTFS drive with execute permission? How can I do that? I'm using R(3.1.1), and ARIMA models for forecasting. I would like to know what should be the &quot;frequency&quot; parameter, which is assigned in the ts() function, if im using time series data which is:  separated by minutes and is spread over 180 days (1440 minutes/day) separated by seconds and is spread over 180 days (86,400 seconds/day).  If I recall right the definition, a &quot;frequency&quot; in ts in R, is the number of observations per &quot;season&quot;. Question part 1: What is the &quot;season&quot; in my case? If the season is &quot;day&quot;, then is the &quot;frequency&quot; for minutes = 1440 and 86,400 for seconds? Question part 2: Could the &quot;frequency&quot;  also depend on what I am trying to achieve/forecast? for example, in my case, I'd like to have a very short-term forecast. One-step ahead of 10minutes each time. Would it then be possible to consider the season as an hour instead of a day? In that case frequency= 60 for minutes, frequency = 3600 for seconds? I've tried for example to use frequency = 60 for the minute data and got better results compared to frequency = 1440 (used fourier see link below by Hyndman)  (The comparison was made by using MAPE for the measure of forecast accuracy) In case the results are complete arbitrary, and the frequency cannot be changed. What would be actually the interpretation of using freq = 60 on my data? I also think it's worth mentioning that my data contains seasonality at every hour and every two hours (by observing the raw data and the Autocorrelation function) How to solve missing libmng.so.1 in Trusty (14.04) Package is no more available I would like to run SoulseekQT. However unfortunately one of the dependencies is libmng1, which is no more included in Trusty. sudo apt-get install libmng.so.1 or libmng1 or libmng1:i386 in Trusty (14.04) no more available. Do you have any idea how to solve this conflict? I mean is it really sure that libmng1 is excluded? Could I use libmng2 somehow instead? If not, could I somehow rebuild libmng1 for Trusty? Or do I have to downgrade back to Saucy (13.10)? I have a series of pending jobs in the queue,   $ atq 8   Mon Oct 29 13:21:00 2018 5   Mon Oct 29 10:36:00 2018 15  Mon Oct 29 13:31:00 2018 4   Mon Oct 29 10:35:00 2018 18  Mon Oct 29 15:55:00 2018 3   Mon Oct 29 10:31:00 2018 16  Mon Oct 29 15:54:00 2018 11  Mon Oct 29 13:24:00 2018 12  Mon Oct 29 13:24:00 2018 17  Mon Oct 29 15:53:00 2018 10  Mon Oct 29 13:23:00 2018 19  Tue Oct 30 15:43:00 2018 2   Mon Oct 29 07:41:00 2018 6   Mon Oct 29 10:41:00 2018 7   Mon Oct 29 13:20:00 2018 9   Mon Oct 29 13:22:00 2018   and plan to remove them job on Oct 29. Fortunately, atrm could remove multiple jobs  atrm 8 5 15 4    despite this, a iteration might be a better solution,   $ atq | grep "Oct 29" 18  Mon Oct 29 15:55:00 2018 3   Mon Oct 29 10:31:00 2018 16  Mon Oct 29 15:54:00 2018 11  Mon Oct 29 13:24:00 2018 12  Mon Oct 29 13:24:00 2018 17  Mon Oct 29 15:53:00 2018 10  Mon Oct 29 13:23:00 2018 2   Mon Oct 29 07:41:00 2018 6   Mon Oct 29 10:41:00 2018 7   Mon Oct 29 13:20:00 2018 9   Mon Oct 29 13:22:00 2018   How could I select the first field of the job number so as to iterating them? GRUB does not detect Windows I can't seem to log in with my Stack Exchange OpenID. When I try, I get the error message:     Unable to log in with your OpenID provider:      The openid.return_to parameter (.) does not match the actual URL (.) the request was made with.   It works if I disable the HTTPS-Everywhere plugin in Firefox, but if I then re-enable the plugin and go to a site on which I am not logged in (in which case login is normally nearly seamless), it fails again with the same error message, or at least something very similar.  I'm not aware of having done anything that might have caused this, although I will admit I don't recall seeing https:// in the address bar on Stack Exchange before (I do now, and that's a good thing IMO!). I was using the site  and it worked fine then, but not now... Create bootable CD from ISO Why two objects get charged by rubbing? How much space does Ubuntu take up after it's already installed? I know how much space is required to install Ubuntu.  How much space does a Ubuntu installation take up on the disk? Can you show me a continuous function $f \colon \mathbb{R}^n\to\mathbb{R}^m$ that satisfies $f(a+b)=f(a)+f(b)$ but is not linear?   We have that  $$f(0)=f(0+0)=2f(0)\implies f(0)=0\\ f(x-x)=f(0)=f(x)+f(-x)=0\implies f(-x)=-f(x)\\ f(nx)=f(x+x+\dots+x)=f(x)+\dots+f(x)=nf(x)\quad \forall n \in \mathbb{N}$$ But $$ f(-nx)=-f(nx)=-nf(x) $$ So: $$ f(ax)=af(x) \quad \forall a \in \mathbb{Z} $$ Plotting an epicycloid Given a set of curved geodesics in 2D space, is there a way to determine the geometrical shape of said space? I have a polygon shapefile of many lakes, I also have a point shapefile of locations on those lakes where fish were caught.  The point shapefile contains the attributes: Fisherman_iD, Fish_Caught, Lat, Long.  The polygon shapefile contains the attributes Lake_name, Points_per_polygon(PNTCNT)  Each point obviously has differing entries under the fish_caught field.   What I want to get is the average fish caught per lake (polygon).  So I need to have a total number of fish caught, per individual polygon, and then I can divide that by the PNTCNT file I create using the Points in a Polygon tool.  Ideas? I am relatively new to QGIS. I'm very new to GIS concepts. Is there a way or procedure of finding out whether the co-ordinate system is in UTM, latitude/longitude or in stateplane using .prj file?   My .prj file contains the following text:      GEOGCS["WGS 84",DATUM["WGS_1984",SPHEROID["WGS   84",6378137,298.257223563,AUTHORITY["EPSG","7030"]],AUTHORITY["EPSG","6326"]],PRIMEM["Greenwich",0,AUTHORITY["EPSG","8901"]],UNIT["degree",0.01745329251994328,AUTHORITY["EPSG","9122"]],AUTHORITY["EPSG","4326"]] Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k&gt;0$ is constant. Show that $f$ is a polynomial of degree at most $n$. Sort thebibliography by citation order How to draw density estimate on histogram by pgfplots? I know how to draw a density histogram using pgfplots from this answer: .  But how can I draw an estimate line on the histogram like this?    \documentclass{article} \usepackage{pgfplots} \begin{document} \pgfplotsset{     small,     every axis plot post/.style={         fill=orange!75,         draw=orange!50!black     },     trim axis left }  \begin{tikzpicture} \begin{axis}[ymin=0, title=\texttt{hist=density}] \addplot [hist=density] table [y index=0] {random.txt}; \end{axis} \end{tikzpicture} \end{document} Check server activity with code I would like to see the activity on of the SQL server. So I can run a query when the server load is low. I know there is an activity monitor in SQL server but my company doesn't want me to give the rights to use this tool. Apparently you can kill other people queries and this is too dangerous.  Is there a way to check the status of the server with code? (Or view the activity without needing those rights so I can manually run them?)