It is well-known that in an integral domain that is not a UFD, elements can have different decomopositions into irreducibles, that do not just differ by units. But in all examples I know, usually only two such decompositions are given. Is it possible for an element to have infinitely many non-equivalent decompositions?
Consider the factorization into irreducibles of $6$ in $\mathbb{Z}[\sqrt{-5}]$. We have $6=2 \times 3$ and  $6=(1+\sqrt{-5}) \times (1-\sqrt{-5})$, i.e. $2$ distinct factorizations. And,  $$6^2=3 \times 3\times2\times2$$ $$=(1+\sqrt{-5}) \times (1-\sqrt{-5}) \times (1+\sqrt{-5}) \times (1-\sqrt{-5})$$ $$=(1+\sqrt{-5}) \times (1-\sqrt{-5})\times3\times2.$$ More generally, $6^n$ will have $n+1$ distinct factorizations into irreducibles in $\mathbb{Z}[\sqrt{-5}]$ by a simple combinatorial argument. But, can we construct a ring in which there exists an element that has an infinite number of distinct factorizations into irreducibles? To make life harder, can we construct an extension of $\mathbb{Z}$ in which this happens? I have been thinking about this for a while and have managed to find no foothold.. Any help is appreciated.
How to approach the modeling of duct work, piping, wiring? Genetically modified humans study insectoid alien miners with super-intelligent hive-mind This is a short story that appeared in a collection. Two genetically-modified humans (corporate anthropologists, basically) travel to a planetoid inhabited by an insect race that is mining the planet in order to study the race. While there, they are practically ignored, as the individual members of the race lack consciousness. The humans determine a type of pheromone is used to control the worker caste, and co-opt several members to create a living area. They make plans to put the mining towards human ends, but before that can happen they are discovered. The presence of the artificial pheromones wakes up a hive-mind, super-intelligent bug, who then bargains with the surviving member of the team (the man) after killing and taking over the female member of the team.  The collection was read at some point between 2000-2005. Bluetooth audio delay / lag, reason completely unclear to me I have a bit of an odd situation, and wanted to ask for ideas.   Recently started connecting a JBL Charge 2+ to my MBA 11 inch, mid-2013, via Bluetooth. I noticed that there's clear delay between sound event and sound being audible via the JBL.   The problems persists across applications (so for example, playing video via VLC means the sound isn't synced, same for youtube videos in different browsers, same for mp3s that start playing with a small delay, etc.)  The delay is minor, but noticeable, in the range of 0.5s I'd say.  At first I thought it might be a Bluetooth driver issues, but now I found out the following: just connected another JBL device, a Flip 3 to the same MBA, and the sound plays without noticeable delay. I'm not sure if they use the exact same drivers, but now I'm wondering:   Is there a chance that it's a hardware problem? Didn't really consider this a possibility, but now I'm not sure anymore. How to test, and, ideally, fix this issue? How can I isolate the reason for the audio lag with the Charge 2+?   Any help would be appreciated! Review audit ignores closereason filter Is there a direct download link with a raw data dump of Stack Overflow? (not a torrent) I would like to create a local MySQL copy of Stack Overflow for academic purposes (school project). As far as I know, the database dump of the  database is available only in an XML format that I will have to convert to SQL on my own with some tool.  Most of direct download links I found were broken (404). The only non-torrent source that I have found and that is still alive is  but the latest Stack&nbsp;Overflow dump there is from October 2009, content of which looks like this:    Where could I find a direct download link with the Stack&nbsp;Overflow dump?    Note: I am not able to use torrents due to network policies, yet for the completeness of this question, here's the link that might help someone:    Other related questions / links that haven't helped me:     ~ all links there give 404  ~ from September '09 with another broken link  ~ from about a year ago with a clearbits.net link  ~ also not working anymore Do Cleric's domain spells add slots? I'm a little bit confused with the whole spell system and additions to them. If I pick for instance Tempest Domain Spell, are the spells additional to the other ones? (2 at level 1) So do that make it 3 spells? Does the domain add slots too?  And also, I've read about people choosing multiple domains, is that possible? A financial planner treated us in a dismissive fashion and I want to say that he "gave us the back of his hand" but that phrase doesn't seem to have the meaning of dismissive. Does it or is there a similar idiom? How do I rename all folders and files to lowercase on Linux? I have to rename a complete folder tree recursively so that no uppercase letter appears anywhere (it's C++ source code, but that shouldn't matter).  Bonus points for ignoring CVS and Subversion version control files/folders. The preferred way would be a shell script, since a shell should be available on any Linux box.  There were some valid arguments about details of the file renaming.   I think files with the same lowercase names should be overwritten; it's the user's problem. When checked out on a case-ignoring file system, it would overwrite the first one with the latter, too. I would consider A-Z characters and transform them to a-z, everything else is just calling for problems (at least with source code). The script would be needed to run a build on a Linux system, so I think changes to CVS or Subversion version control files should be omitted. After all, it's just a scratch checkout. Maybe an "export" is more appropriate. My public website name and AD domain name are the same. How can I get to my external website from inside my network? I use pgfplots for most of my plots. Now I need to add a . Is this possible with pgfplots or any other package? Can I list-initialize a vector of move-only type? What insect is this? (India) Can anyone tell me what kind of insect (if it is one) is this...!! Or is it the pupa of some insect?  Well I don't think that the whole big thing is the insect itself, it appears to be just a kind of protection or shelter which moves along with the tiny little thing that keeps on popping out and going it so as to move. Location:  India Where can I find a repository for older unsupported releases? Indian CGPA to US GPA system How can I switch a 32-bit installation to a 64-bit one? I've got a 32-bit Ubuntu installation running on 64-bit hardware. Now that , I would like to switch to 64-bit without having to reinstall the OS.  This is one of the :  Shawn installed his system using the 32-bit version of Ubuntu, but his hardware is 64-bit and he wants to switch over. He manually installs the amd64 versions of dpkg and apt, replacing the i386 versions and changing which architecture is used as the default; then he installs the amd64 ubuntu-minimal package; then he installs the amd64 ubuntu-desktop package. Over time the remaining i386 packages are replaced automatically on upgrade.  However, when trying to follow the instructions in there, I cannot find any 64-bit version of dpkg or apt.  Did this user story got implemented in a different way in the final spec, or do I need to do something differently?  In short, how can I switch my 32-bit installation to 64-bit? Consider the diagram below:  (Author: Joshua Hykes )  From this diagram we can see that the absorption coefficient for the photoelectric effect generically decreases with the increase in energy of the photon. What is the physical reason for this decrease. (i.e. why does the probability of photoelectric absorption decreases with increasing photon energy).  Note: I am aware that the large spin is due to more inner electrons been able to be lifted.  Edit For Clarity  I will here rephrase the question: in the above diagram there is a general decrease in the absorption coefficient due to the photoelectric effect (i.e. the absorption of a photon by an electron and subsequent emission of that electron) with energy (ignoring the occasional absorption edge). This is of course linked to a decrease in the probability of the photoelectric effect occurring (and a corresponding decrease in the cross section). My question is: what is causing this decrease with increasing energy. (For the purpose of this question I am ignoring the contribution due to the Compton effect and pair production and simply focusing on the cross section due to the photoelectric effect.) Parse HTML in Android Clarification of the "25D" option in ogr2ogr When is matrix multiplication commutative? I know that matrix multiplication in general is not commutative. So, in general:  $A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$  But for some matrices, this equations holds, e.g. A = Identity or A = Null-matrix $\forall B \in \mathbb{R}^{n \times n}$.  I think I remember that a group of special matrices (was it $O(n)$, the ?) exist, for which matrix multiplication is commutative.  For which matrices $A, B \in \mathbb{R}^{n \times n}$ is $A\cdot B = B \cdot A$? Is circular reasoning always a fallacy?