In which rings $R$ does $a\mid xy$ imply the existence of $b\mid x$ and $c\mid y$ with $a=bc$? Let $R$ be some integral domain and consider the following property: For all $a,x,y\in R$, if $a\mid xy$, then there exist $b\mid x$ and $c\mid y$ with $a=bc$.  Question: Is there a 'simple' classification of integral domains with this property?    Own work: Note that the property implies that irreducible elements are prime, so if $R$ satisfies the ascending chain condition on principal ideals, it must be a UFD. Since the property clearly holds in any UFD, the only 'interesting' $R$ are those that satisfy the property, but not the ascending chain condition on principal ideals.   Perhaps one can prove that the property is equivalent with irreducible elements are prime, in which case it holds precisely when $R$ is a GCD domain. This would be the type of 'simple' classification I'm looking for.
In what kind of rings a divisor of a product is a product of divisors? In a , if $a|bc$, then $a$ can be written as $a = a_1a_2$ so that $a_1|b$ and $a_2|c$.  Is this property of a commutative ring strictly weaker than the property of being a UFD?  If so, does it have a name, or is it equivalent to some other well-known property?
Just a few questions about Hyper-V virtual machines and clustering I have been using Microsoft’s Hyper-V technology for a little while now, but I am just now dipping my toe into clustering.  In particular, I am trying to implement a fault-tolerant SQL DB. This involves setting up two VMs, clustering them via Failover Cluster, and then installing SQL Server in some fashion. I have two physical machines - one high-end and rather beefy “heavy lifter” to contain the majority of the VMs, and another “backup” (a repurposed desktop) to hold the essential “secondary” (or failover) AD-DC, SQL and FS VMs.  The main reason why I find the failover cluster at the VM level so attractive is that it presents a single IP and DNS entry to the network as a whole - if one machine (physical or virtual) goes down, you might loose some ping and the connections get reset, but the network applications (Microsoft RMS connection to backend SQL) can still connect to a viable DB without having to mess around with the settings at all.  My first question is in terms of SQL Server itself. If I have a cluster between two VMs, does it make more sense to install the SQL Server in Failover Cluster configuration or should I simply install it in a stand-alone config and mirror the DBs? For example,  suggests just mirroring the DBs, but do I just mirror standalone DBs on standalone VMs, or can I get the network and failover benefits of clustered VMs while still utilizing (on each clustered VM) standalone DBs that have been mirrored between each other?  As well, I have come across a lot of documentation about SQL clustering, but most assume a number (#>2) of physical machines to hold not only the actual SQL VMs but also the Quorum and Witness stores. I will not be able to muster more than two physical machines. As such, I will have to be satisfied with a VM cluster that does not exceed two VMs (one for each physical machine).  Another issue involves MSDTC - the Distributed Transaction Coordinator. When attempting to install the SQL Failover Cluster (I never completed it for this reason) it threw a hissy fit because MSDTC had not been clustered. Search as I might, I have not yet found a way to do so under Windows Server 2012 R2. I have found , but these instructions don’t align with 2012 R2 (at least, not in a way that allows me to successfully cluster MSDTC). Plus,  that I have found for SQL Server Failover Cluster installation suggest that a third “network device” - shared network storage (a SAN) - is required for the DB itself (and other functionality). I do not have this, and won’t be getting this. Most of my storage exists on the “heavy lifter” that was designed for all of the “primary” VMs. If that physical machine goes down, so does the storage. The secondary server does have enough resources for an AD-DC Server, an SQL server and a File Server, so it will handle the “secondary” failover versions of those VMs (clustered or not).  My final question involves file servers. If I cluster file servers between two VMs (one on my “heavy lifter” and another on my “backup”, how do I mirror the data between them? Clustering VMs only provides a single point of access on the network for a resource, it doesn’t exactly replicate data between the two - that is left to the services that serve up that data. I am unsure how I can ensure that file server data between two clustered file server VMs can be properly mirrored. Remember, I only have two devices to be used here - my primary machine and a backup secondary. There is no chance of me obtaining a SAN or any other type of network attached storage. What exists on the machines must act as the storage.  Thanks in advance for any suggestions. I am working on refactoring some old code and have found few structs containing zero length arrays (below). Warnings depressed by pragma, of course, but I've failed to create by "new" structures containing such structures (error 2233). Array 'byData' used as pointer, but why not to use pointer instead? or array of length 1? And of course, no comments were added to make me enjoy the process...  Any causes to use such thing? Any advice in refactoring those?  struct someData {    int nData;    BYTE byData[0]; }   NB It's C++, Windows XP, VS 2003 QThread and QTimer This question is regarding the minecraft Beta 1.8 prerelease -- I have a beautiful house in single player survival which I'd like to record while flying.  I want to be able to change the game mode of my survival map to creative to be able to fly through it without installing a mod. I know it is possible if you run the map on a multiplayer server via the command /gamemode &lt;user&gt; 0|1 but I want to achieve this in single player :)  I think this would be possible through editing your level.dat file as I think that's where the game mode for the map is stored. Generic htaccess redirect www to non-www I would like to redirect www.example.com to example.com. The following htaccess code makes this happen:  RewriteCond %{HTTP_HOST} ^www\.example\.com [NC] RewriteRule ^(.*)$ http://example.com/$1 [L,R=301]   But, is there a way to do this in a generic fashion without hardcoding the domain name? Is division of matrices possible? Is it possible to divide a matrix by another? If yes, What will be the result of $\dfrac AB$ if $$         A = \begin{pmatrix}         a &amp; b \\         c &amp; d \\         \end{pmatrix},         B = \begin{pmatrix}         w &amp; x \\         y &amp; z \\         \end{pmatrix}? $$ I'm at roughly 67% completion of the Alola Dex, with ~200 caught, ~290 seen. After scanning QR codes, I've caught a Swinub and Cyndaquil. However, neither were registered in the Pokédex, since neither are Pokémon in the Alola Dex. This brought up the following question:  When will I unlock the National Dex?   Do I need to have seen all the Pokémon in the Alola Dex? Do I need to complete some specific quest, like catching/defeating the Ultra Beasts? I am running SQL Server 2012.  The SQL Server Management Studio has the option to right click on a database then select Tasks and Generate Scripts.  Is there a way to automate that via command line somehow?  I want to create a script that includes the schema and data of the entire data base.  Tools like ScriptDB and sqlpubwiz.exe all seem to target SQL Server 2005. What about SQL Server 2012? Using the same secret key for encryption and authentication in a Encrypt-then-MAC scheme Is it a weakness to use a single shared secret for protecting messages using a Encrypt-then-MAC scheme?    Assuming a system is using AES-256-CBC and a SHA1-HMAC and the same secret key for both operations.  Upon intercepting one of these messages and attacker could brute force this secret key by trying to solve the HMAC or by successfully decrypting the first block of the ciphertext.  Performing  a decryption on a single block of AES-256 should be a lighter operation than performing a SHA1-HMAC on a potentially large message.  Therefore AES-256 is ideal for a massively parallel brute-force attack and the use of the SHA1-HMAC doesn't hurt the system.  Is this approach sound, or is there an attack that has not been taken into consideration? I'm a C++ newbie, but I wasn't able to find the answer to this (most likely trivial) question online.  I am having some trouble compiling some code where two classes include each other.  To begin, should my #include statements go inside or outside of my macros?  In practice, this hasn't seemed to matter.  However, in this particular case, I am having trouble.  Putting the #include statements outside of the macros causes the compiler to recurse and gives me "#include nested too deeply" errors.  This seems to makes sense to me since neither class has been fully defined before #include has been invoked.  However, strangely, when I try to put them inside, I am unable to declare a type of one of the classes, for it is not recognized.  Here is, in essence, what I'm trying to compile:  A.h  #ifndef A_H_ #define A_H_  #include "B.h"  class A {     private:         B b;      public:         A() : b(*this) {} };  #endif /*A_H_*/   B.h  #ifndef B_H_ #define B_H_  #include "A.h"  class B {     private:             A&amp; a;      public:         B(A&amp; a) : a(a) {}  };  #endif /*B_H_*/   main.cpp  #include "A.h"  int main() {     A a; }   If it makes a difference, I am using g++ 4.3.2.  And just to be clear, in general, where should #include statements go?  I have always seen them go outside of the macros, but the scenario I described clearly seems to break this principle.  Thanks to any helpers in advance!  Please allow me to clarify my intent if I have made any silly mistakes! Ignored users still appear in the 'other rooms you're in' message preview Well it might not be a huge issue to some (I suppose it depends on how you use the chat system) but when I "Ignore this user everywhere", their messages still appear on the sidebar for the 'other rooms you're in' if they say something. Albeit I can ignore this feature and the text within it, I just find it odd that I can still see it.   This is only a nuisance when you truly are annoyed by a user and ignoring them in the chat was the best possible alternative to leaving the room or starting an argument (no references here, but any long time chat user has been annoyed by some help vampire at one point in time or another and would just prefer to ignore them for good).   If you have no clue to the sidebar or preview I'm talking about: this is for you,    I have chosen to Ignore this user everywhere (Neal) and then switched to a new room, yet that users response appears in my 'preview' for the 'other rooms you're in' if he is also talking in them.  Note: My ignore of Neal was mostly due to jest, this in no way is to reflect on Neal but the chat feature (I hate to ever single anyone out here). Okay, obviously beyond Data and the Soong-type androids. But seriously, we have more robots in our daily life now than there apparently are on Trek. Is there a canon answer why not? It seems like Star Wars is more on the nose in this regard - it’s unclear why all dirty/dangerous/boring work wouldn’t be done by robots (anthropomorphic or not). How to understand the notion of a differential of a function I get      every few minutes on Kubuntu (KDE), and the panels disappear.  It restarts itself automatically, so it's not a real problem, just an annoyance, but it is a big annoyance.  I've tried sudo apt install --reinstall plasmashell plasma-desktop, but that made no difference.  When I try a backtrace from the error message, it says it's unable to produce a useful backtrace. How do I create a file of size 10M, format it with ext3 filesystem and then mount it in /media/fuse?  I tried with  mkfs -t ext3 file   then to mount it  mount -t ext3 file /media/fuse   It didn't work because it said that file wasn't a block device. Can anybody help me? What's the difference between  and ,  and ? What's the difference between &lt;b&gt; and &lt;strong&gt;, &lt;i&gt; and &lt;em&gt; in HTML/XHTML? When should you use each? Is there a tool to get the lines in one file that are not in another? Is there any tool that can get lines which file A contains, but file B doesn't? I could make a little simple script with, e.g, perl, but if something like that already exists, I'll save my time from now on. 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 $(X_1,...,X_n)$. Why am I getting "size mismatch" errors from the Iceland mirror? After installing Ubuntu 12.04 (clean install) I fetch a bunch of program via synaptic (installed specially), all programs worked except Gimp 2.8. So I tried purging Gimp, install again (no luck). Checked broken packages and reinstalled them. When using the Software center I get this MSG:  Failed to fetch http://is.archive.ubuntu.com/ubuntu/pool/main/o/openexr/libopenexr6_1.6.1-4.1_amd64.deb Size mismatch   Tried also this:  sudo apt-get purge gimp* sudo add-apt-repository ppa:otto-kesselgulasch/gimp sudo apt-get update sudo apt-get install gimp   And get this MSG:  Failed to fetch http://is.archive.ubuntu.com/ubuntu/pool/main/o/openexr/libopenexr6_1.6.1-4.1_amd64.deb  Size mismatch E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing?   I am wondering if the Icelandic Archive has an outdated version or something?   would really like to install Gimp 2.8 but Gimp 2.6 is OK to.   I am also wondering if I am doing something wrong? Prevent direct access to a php include file I have a php file which I will be using as exclusively as an include. Therefore I would like to throw an error instead of executing it when it's accessed directly by typing in the URL instead of being included.  Basically I need to do a check as follows in the php file:  if ( $REQUEST_URL == $URL_OF_CURRENT_PAGE ) die ("Direct access not premitted");   Is there an easy way to do this?