How would I prove there is no unique factorisation in $\mathbb Z[\sqrt{d}]$ for $d \leq-3$, where $d$ is a square-free integer?   I think it's something to do with the only invertible elements being $\pm 1$
I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD.   I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$ are all irreducible. Is there someway to conclude that $\mathbb{Z}[\sqrt{-n}]$ is not a UFD based on this? Thanks.
How can two dates be compared in a shell?  Here is an example of how I would like to use this, though it does not work as-is:  todate=2013-07-18 cond=2013-07-15  if [ $todate -ge $cond ]; then     break fi                              How can I achieve the desired result? Can not update Bash on Debian 6.0 (Squeeze) I can not update  on a Debian 6.0 (Squeeze) server to get rid of the discovered vulnerability:  bash --version GNU bash, version 4.1.5(1)-release (x86_64-pc-linux-gnu)  apt-get update apt-get install bash Reading package lists... Done Building dependency tree Reading state information... Done bash is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 7 not upgraded.   Can I use Squeeze-LTS for this server just to update Bash? After one week I will be on another server, so I will not make any other updates.  uname -m x86_64  lsb_release -a No LSB modules are available. Distributor ID: Debian Description:    Debian GNU/Linux 6.0.5 (squeeze) Release:        6.0.5 Codename:       squeeze TSA requirements regarding ID If the name on my boarding pass lists Debbie and my Drivers License lists Debra, will this be a problem? How To Get Command Blocks Talking in Chat I need a way for a command block to say things in chat when it is powered, or more preferably whenever someone spawns into the world. I'm in a numerical analysis course right now and it's pretty rigorous but I'm enjoying it a lot. I took a lower level course before that was more oriented towards implementation of numerical methods, so it's not my first time with the material.  Anyway, my professor wrote his own course notes and they're fantastic but I wanted some reference material since I'm currently outlining my own notes. Just wondering what the big ones out there are  edit: I would say I'm at an advanced undergraduate level Where can I get Blender test builds? I know that  provides downloads for official releases. I see on forums that people use newer versions of Blender. Where can I get newer version of Blender (nightly builds, alpha versions... etc). When is the completion of a ring a local ring ? Let $R$ be a commutative ring with unit and let $m$ be a maximal ideal of $R$. Are there known conditions on $R$ or $m$ such that the $m$-adic completion $\hat{R}$ of $R$ is a local ring.   Since the completion of a Noetherian local ring is again local, I'm primarily interested in cases where $R$ itself is not local.   An example is the polynomial ring $R=k[X_1,...,X_n]$ ($k$ a field) with $m=(X_1,...,X_n)$ where the power series ring $\hat{R}=k[[X_1,...,X_n]]$ is local with maximal ideal $(X_1,...,X_n)$. I have seen many examples of how to render sprites from a spritesheet but I havent grasped why it is the most common way of dealing with sprites in 2d games.  I have started out with 2d sprite rendering in the few demo applications I've made by dealing with each animation frame for any given sprite type as its own texture - and this collection of textures is stored in a dictionary. This seems to work for me, and suits my workflow pretty well, as I tend to make my animations as gif/mng files and then extract the frames to individual pngs.  Is there a noticeable performance advantage to rendering from a single sheet rather than from individual textures? With modern hardware that is capable of drawing millions of polygons to the screen a hundred times a second, does it even matter for my 2d games which just deal with a few dozen 50x100px rectangles?  The implementation details of loading a texture into graphics memory and displaying it in XNA seems pretty abstracted. All I know is that textures are bound to the graphics device when they are loaded, then during the game loop, the textures get rendered in batches. So it's not clear to me whether my choice affects performance.  I suspect that there are some very good reasons most 2d game developers seem to be using them, I just don't understand why. For privacy and performance reasons, I'd like to prevent my computer from ever communicating with certain Internet hosts. I'd like to do this at the OS level, not through web browser plug-ins (not broad enough), or even through my home router (my laptop travels). I'd prefer to use built-in tools.  I'm running Windows 7 Professional 64-bit. I'm all too familiar with the Windows hosts file - and its limitations. There seems to be some confusion over whether hosts supports wildcards, but my own tests confirm that it does not.  I have experimented with Windows Firewall and the IP Security Policy MMC snap-in. So far as I can tell, both require numeric addresses. I don't want to block specific IP addresses, or ranges of addresses, in part because IP-address-to-hostname mappings can and do change.  Is there any tool in Windows 7 Professional with which I can block (or route to 0.0.0.0) communication with Internet hosts by hostname mask? If not, are there any good, free, third-party tools? I installed Ubuntu on a Dell XPS 13 laptop that had Windows 10 preinstalled on it. I installed it from a USB flash drive on a partition. When I boot my computer I can only boot into Windows 10 and Ubuntu is nowhere to be seen.  If I boot into my USB then I can see that Ubuntu is installed, but I can't get to it from the BIOS boot menu. Why cannot fermions have non-zero vacuum expectation value? In quantum field theory, scalar can take non-zero vacuum expectation value (vev). And this way they break symmetry of the Lagrangian. Now my question is what will happen if the fermions in the theory take non-zero vacuum expectation value? What forbids fermions to take vevs? How does chain fishing work? We are getting  in M19. What if I enchant my , copying my opponents . Then I pay the cost on my Bolas (previously a Delver) to activate his transform ability.  What will happen? I can think of 3 possibilities:   My Delver is exiled and the aura goes to the graveyard. My Delver is transformed, aura goes to the graveyard, but the Delver transforms into  My Delver is transformed, aura goes to the graveyard, and the Delver transforms into Share question to Social Media option or button is much needed Few days back while attending a thesis defense, one Professor was asking the defender why does he think, he deserves the degree. I was wondering since then, is there really any general answer for that? or the answer should be given describing my own work and then emphasis them how it is important for research field?  I asked the question here to know your opinion about it and to list down what could be the possible answers to this question. Inline math equation Point size How to change all equations point size(inline and display equations) globally, point size worked in within \begin{array}...\end{array} but not changed for $....$?  Please find MWE file:  \documentclass{book} \usepackage{amsmath,array} \usepackage{stix}  \everydisplay{\fontsize{15pt}{18pt}\selectfont} \everymath{\fontsize{15pt}{18pt}\selectfont}  \parindent0pt \begin{document}  Use the alternative calculation described above. In comparing the two values, you want to consider fluctuations in inventory over time. Use the alternative calculation described above. In comparing the two values, you want to consider fluctuations in inventory over time.  $\alpha\beta\gamma\delta\epsilon0123456789+-$ \bigskip  $\begin{array}{@{}l}\alpha\beta\gamma\delta\epsilon0123456789+-\end{array}$   \begin{align*} \alpha\beta\gamma\delta\epsilon0123456789+- \end{align*}  \end{document} some more examples:   "And she gave me that aren't-I-just-gorgeous smile." "The I-did-it-my-way approach." "A from-this-day-forward-I-have-no-son scene." I have the amulet of Mara but when I go to talk to maramal about a wedding nothing shows up. WHY is that. How does the radius of a pipe affect the rate of flow of fluid? Poiseuille's law states that the rate of flow of water is proportionate to $r^4$ where $r$ is the radius of the pipe. I don't see why.  Intuitively I would expect rate of flow of fluid to vary with $r^2$ as the volume of a cylinder varies with $r^2$ for a constant length. The volume that flows past a point is equal to rate of flow fluid, and thus it should vary with $r^2$.  I cannot understand the mathematical proof completely. Is there an intuitive explanation for this? Is there a geometric reason why two merging black holes never “decay” into two separate black holes All black hole mergers seem to result in one Kerr black hole after ring-down.   Could it be possible, that two initial black holes with two separate event horizons form an “intermediate compound state” with one event horizon, which - at a later time - “decays” in two final black holes?  I know that the answer to this question is “no” when applying Bekenstein-Hawking entropy where the decay would violate the second law which postulates $dA/dt &gt; 0$.   And the answer seems to be “no” when relying on numerical relativity for black hole mergers.  My question is if there is a more elegant, geometrical proof following entirely from general relativity, which rules out such a decay.  Thanks