Since   $ G =( \prod_{1}^{\infty} \mathbb{Z} )\times  \mathbb{Z}  \times \mathbb{Z}  \cong (\prod_{1}^{\infty} \mathbb{Z}) \times \mathbb{Z}  $,   by taking quotients we get  $\mathbb{Z \times Z} \cong G/ \prod_{1}^{\infty} \mathbb{Z} \cong \mathbb{Z}$.  Therefore   $\mathbb{Z} \cong \mathbb{Z \times Z}$.  But $\mathbb{Z}$ is indecomposable!  What's wrong with above proof?
Let $G$, $H$, and $K$ be abelian groups. Assume $G \oplus H \cong G \oplus K$. If each are finitely generated, then the structure theorem easily gives us that $H \cong K$. However, there are counterexamples in the general case, namely $H=\mathbb{Z}$, $K=\mathbb{Z} \oplus \mathbb{Z}$, and $G=\bigoplus_{i=1}^\infty\mathbb{Z}$.   I am wondering what goes wrong with the following argument:   Form the split short exact sequence  $$0 \to G \xrightarrow{i} G \oplus H \xrightarrow{\pi} H \to 0$$  where $i$ and $\pi$ are the inclusion and projection, respectively. Then don't we immediately get from the first isomorphism theorem that $H \cong (G \oplus H)/G$, which would contradict the counterexample above?   If we run this for the counterexample given, we get the sequence   $$0 \to \bigoplus_{i=1}^\infty\mathbb{Z} \to \bigoplus_{i=1}^\infty\mathbb{Z} \to \mathbb{Z} \to 0$$  which seems to still be short exact, even though here $G \cong G \oplus H$. But of course here we can see directly that $(G \oplus H)/G \cong G/G$, which is trivial. So what went wrong?   It seems to me to be something along these lines: In this case, the kernel of the projection is not the same as the original group, it is just, say, $0 \oplus \mathbb{Z} \oplus \mathbb{Z} \oplus \dots$, i.e. nontrivial from the second coordinate on, which just happens to be isomorphic to the original group. So if we form the other short exact sequence with $K$ in place of $H$, then the difference seems to be that we have two different projection maps with different (but isomorphic) kernels.   But I'm not really sure how to explain this in the general case, if my reasoning is correct to begin with.   Thanks in advance for any help!
I have a DVD I was trying to rip, and handbrake froze up when scanning it.  I tried to eject the DVD by dragging and dropping it to the trash, which trashed the icon, but didn't eject the DVD.  I then forced Handbrake to quit, but without an icon on my desktop to select, how can I eject the DVD? I have a site that is in French and English. I am able to successfully setup so that users can switch back and forth between the two domains to view either language, however I cannot seem to keep users logged in. When they are logged into the english domain and then switch to the french domain they are required to login again. However if they switch back to the english domain they are still logged in.  As an example we'll use www.english-domain.com as the base (default) domain and www.french-domain.com as the secondary domain  In settings.php I have modified the following, however it doesn't seem to be keeping them logged in. Am I missing something?  /**  * Drupal automatically generates a unique session cookie name for each site  * based on its full domain name. If you have multiple domains pointing at the  * same Drupal site, you can either redirect them all to a single domain (see  * comment in .htaccess), or uncomment the line below and specify their shared  * base domain. Doing so assures that users remain logged in as they cross  * between your various domains.  */ $cookie_domain = 'english-domain.com';   Note: That I have also flushed both site and browser cache as well as deleted any cookies left on my machine.  Update:  Here are my language settings: How to import Apple Mail archive files in Outlook 2013? I need to import Apple Mail archived files in Outlook 2013 on Windows, but every time I open the import wizard and reach the file selection step, I cannot select the archived file. It doesn't even show in the dialog box.  I don't know what the issue is. Is it the file size issue as the archive file is large (about 7 GB) or there is any other fault? C# getting its own class name Using continuous color band for raster legend in QGIS composer legend? I'm looking for a way to insert continuous colour legend for a raster layer.  I managed to create discrete symbols with values along the bar, but I would like to have a continuous bar with some values on the bar, similar to the old plugin "One-Band Raster", that it doesn't work with the QGIS version higher than 1.7  I tried to set in properties->style, where I selected the  Render Type "Singleband pseudocolor" and see the "generate new color ramp" box.  I tried both "Continuous" or "Equal interval". However, in the Composer the legend always is displayed with discrete symbols.  Is there any way that can make QGIS Composer to display a nice continuous band for a raster layer? Batch Rename (remove specific character) Files How to clip an Openlayers OSM Background to a Vector Boundary? I'm new to QGIS. In my project i have two layers: one is my vector (city boundaries) and the other is the OSM (obtained with the plugin). How can I cut the OSM along the edges of the vector? I want to obtain only a region of the map, to save it as an image. skype 4.3 cannot connect to account on ubuntu 14.04 LTS I have being using Skype for a long time on my Ubuntu 14.04 LTS. From this morning on, Skype simple cannot connect to my account anymore.  That is my Skype version:     Skype 4.3.0.37   © 2014 Skype and/or Microsoft   My Ubuntu system:  LSB Version:    core-2.0-amd64:core-2.0-noarch:core-3.0-amd64:core-3.0-noarch:core-3.1-amd64:core-3.1-noarch:core-3.2-amd64:core-3.2-noarch:core-4.0-amd64:core-4.0-noarch:core-4.1-amd64:core-4.1-noarch:cxx-3.0-amd64:cxx-3.0-noarch:cxx-3.1-amd64:cxx-3.1-noarch:cxx-3.2-amd64:cxx-3.2-noarch:cxx-4.0-amd64:cxx-4.0-noarch:cxx-4.1-amd64:cxx-4.1-noarch:desktop-3.1-amd64:desktop-3.1-noarch:desktop-3.2-amd64:desktop-3.2-noarch:desktop-4.0-amd64:desktop-4.0-noarch:desktop-4.1-amd64:desktop-4.1-noarch:graphics-2.0-amd64:graphics-2.0-noarch:graphics-3.0-amd64:graphics-3.0-noarch:graphics-3.1-amd64:graphics-3.1-noarch:graphics-3.2-amd64:graphics-3.2-noarch:graphics-4.0-amd64:graphics-4.0-noarch:graphics-4.1-amd64:graphics-4.1-noarch:languages-3.2-amd64:languages-3.2-noarch:languages-4.0-amd64:languages-4.0-noarch:languages-4.1-amd64:languages-4.1-noarch:multimedia-3.2-amd64:multimedia-3.2-noarch:multimedia-4.0-amd64:multimedia-4.0-noarch:multimedia-4.1-amd64:multimedia-4.1-noarch:printing-3.2-amd64:printing-3.2-noarch:printing-4.0-amd64:printing-4.0-noarch:printing-4.1-amd64:printing-4.1-noarch:qt4-3.1-amd64:qt4-3.1-noarch:security-4.0-amd64:security-4.0-noarch:security-4.1-amd64:security-4.1-noarch Distributor ID: Ubuntu Description:    Ubuntu 14.04.3 LTS Release:    14.04 Codename:   trusty   Any suggestion why or how to fix it?  Thanks in advance. I use a lot of lists and arrays but I have yet to come across a scenario in which the array list couldn't be used just as easily as, if not easier than, the linked list.  I was hoping someone could give me some examples of when the linked list is notably better. When can we speak of collinearity In linear models we need to check if a relationship exists among the explanatory variables. If they correlate too much then there is collinearity (i.e., the variables partly explain each other). I am currently just looking at the pairwise correlation between each of the explanatory variables.  Question 1: What classifies as too much correlation? For example, is a Pearson correlation of 0.5 too much?  Question 2: Can we fully determine whether there is collinearity between two variables based on the correlation coefficient or does it depend on other factors?  Question 3: Does a graphical check of the scatterplot of the two variables add anything to what the correlation coefficient indicates? Return a blank when cell reference is blank I am using the following formula in a Google Sheets to calculate the number of days between two dates (dates are in columns G and AL):  =ARRAYFORMULA(if(row(A:A)=1, "Readmission Days", (G:G)-(AL:AL)))   It has to be an array because I am working with a Google Forms and entries are automatically populated into new rows.  The problem is that some of the cells in col AL are blank, so the formula is reading those dates as something else and returning a very large number. Also, when both cells are blank, the formula returns a 0, messing up my data.  How can I include something to ignore blank cells or just return a blank when one or both cell references is blank? Swapping right shift and delete key using xmodmap I'm unsure how to swap these 2, this is the thought process I had but am unsure how to get it working:  remove sh = Shift_R remove dl = Delete  keysym Delete = Shift_R keysym Shift_R = Delete  add sh = Delete add dl = Shift_R   This is based on the code I use to swap caps lock and control.  Any thoughts?  xmodmap output:  dan@lantea:~$ xmodmap -pke | grep Shift_R keycode  91 = Shift_R NoSymbol Shift_R keycode 119 = Shift_R NoSymbol Shift_R dan@lantea:~$ xmodmap -pke | grep Delete keycode  62 = Delete NoSymbol Delete Let's have breakfast    Let's have a good breakfast      I've learned that indefinite articles aren't used before uncountable nouns, so why is a used before the adjective "good" here?    Is there an exception to the rules or is it wrong? Find expected value of $\frac{1}{1+\xi}$ Find expected value of $\frac{1}{1+\xi}$ if $\xi$ has Poisson distribution.  My try:  $$E\left[ \frac{1}{1+\xi}\right]=\sum_{k=0}^{\infty}{x}_{k}{p}_{k}=\sum_{k=0}^{\infty}\frac{1}{k+1}\frac{\lambda^ke^{-\lambda}}{k!}$$  Is it right? How to get horizontal black line when coloring cells? Chain skip happening seemingly at random with a new chain 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! raw_input in python without pressing enter How do I Uninstall Windows 10 or Windows 10 preview? When should I use the "strictfp" keyword in java? I've looked up what this does, but does anyone actually have an example of when you would use the strictfp keyword in Java? Has anyone actually found a use for this?  Would there be any side-effects of just putting it on all my floating point operations?