Subgroup generated by $\sqrt2$ is dense in $[0,1]$.
Why is this quotient space not Hausdorff?
In iOS 7, when the phone is in Sleep mode and I receive a call, I don't see the "Decline" button. All I see is a green "Answer..." button that requires me to slide to the right to answer the call. Is there anyway we can change that? I know that there are buttons for "Remind Me Later, etc" but I want the "Decline" button.   It works fine when the phone is not in Sleep mode. Can I install Microsoft Office? If so, how? How to get correlation between two categorical variable and a categorical variable and continuous variable? I am building a regression model and I need to calculate the below to check for correlations   Correlation between 2 Multi level categorical variables Correlation between a Multi level categorical variable and continuous variable  VIF(variance inflation factor) for a Multi level categorical variables   I believe its wrong to use Pearson correlation coefficient for the above scenarios because Pearson only works for 2 continuous variables.   Please answer the below questions   Which correlation coefficient works best for the above cases ?  VIF calculation only works for continuous data so what is the alternative?  What are the assumptions I need to check before I use the correlation coefficient you suggest?  How to implement them in SAS &amp; R? Where to put the periods when using a parenthetical sentence? I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? Indentation shortcuts in Visual Studio How do I start with in-camera JPEG settings in Lightroom? I'm a novice shooting in RAW and find that half the time my camera JPEG images come out better than my Lightroom tweaks to RAW files.  So to give myself a better start I'd like to apply the JPEG settings that are applied automatically by the camera to my RAW files in lightroom so that initially my RAW files look exactly like the JPEG files.   How can I do this? Partition of circumference into $3k$ arcs The following problem is from 1982 Russian Mathematical Olympiad. If you go to this , and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file.     Let $k$ be a positive integer. There are $3k$ points marked on the circumference. They divide the circumference onto $3k$ arcs. Some $k$ of arcs have length 1, other $k$ of them have length 2, the rest $k$ of them have length 3. Prove that some two of the marked points are the ends of one diameter.   I haven't given too much thought into the problem yet (I probably wouldn't get anywhere), but I thought I would share this awesome problem.   Added. Trivial Observation: The length of the circumference is $k+2k+3k=6k$. How do I time a method's execution in Java? . (See comments to the question: the OP tried to include a couple of long XML files, but that wasn't possible.)  What's the recommended approach to including code/data, when there's just too much of it to embed in the question?  I could think of a few approaches myself:   Paste the data to a pastebin and link (and wait for the links to die) Encode the (compressed) data in an image and include that (kludgy!) Implement an attachment feature in Stack&nbsp;Overflow (which will probably be abused) How do I pipe each command given to the shell? I'd like to edit my .bashrc so that every command executed on the shell is piped to something, for example:   $ sudo apt update   _________________ &lt; sudo apt update &gt;  -----------------     \   ^__^      \  (oo)\_______         (__)\       )\/\             ||----w |             ||     ||   I have managed something rather similar, but not entirely:  $ bash $ exec &gt; &gt;(cowsay) $ echo "Hello AU!" $ exit  _______ &lt; Hello AU! &gt;  -------     \   ^__^      \  (oo)\_______         (__)\       )\/\             ||----w |             ||     ||   It is not the desired result, as it only happens after exiting the current shell.  It's mainly for fun/learning purposes. What is a word for someone who is experienced and wise, but who deliberately acts naïve?    I don’t intend it for sarcastic use; I’m trying to describe someone succeeding in making people think that that person truly is naïve.   Is there a word for this? How to create an index map of atlas features in QGIS 3? I realise that this question has been asked before (), however I am still unable to figure out how to create an index map of atlas features in the QGIS map composer.  Basically, I have a line and I have five atlas maps showing the extent of the line, and to show where which map corresponds to the line a I need five overviews on a map with numbers on them, which will look similar to below:    I've created five overviews, but as all of these must be set to correspond to the main map frame, all of them show the overview for which ever atlas feature is selected at the time, rather than appearing as separate index tiles.  Can anyone tell me where I am going wrong? I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant will tell about the volume of the space enclosed by those vectors. Does trace give any information about the space? Rank of the $n \times n$ matrix with ones on the main diagonal and $a$ off the main diagonal I want to find the rank of this $n\times n$ matrix \begin{pmatrix} 1 &amp; a &amp; a &amp; \cdots &amp; \cdots &amp; a \\ a &amp; 1 &amp; a &amp; \cdots &amp; \cdots &amp; a \\ a &amp; a &amp; 1 &amp; a &amp; \cdots &amp; a \\ \vdots &amp; \vdots &amp; a&amp; \ddots &amp; &amp; \vdots\\ \vdots &amp; \vdots &amp; \vdots &amp; &amp; \ddots &amp; \vdots \\ a &amp; a &amp; a &amp; \cdots  &amp;\cdots &amp; 1 \end{pmatrix}  that is, the matrix whose diagonals are $1's$ and $a$ otherwise, where $a$ is any real number.  My first observation is when $a=0$ the rank is $n$ and when $a=1$ the rank is $1.$ Then I can assume $a\neq 0, 1$ and proceed row reduction to find its pivot rows. I obtain  \begin{pmatrix} 1 &amp; a &amp; a &amp; \cdots  &amp; a \\ 0 &amp; 1+a &amp; a &amp; \cdots &amp; a \\ 0 &amp; a &amp; 1+a &amp; \cdots &amp; a \\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ 0 &amp; a &amp; a &amp; \cdots  &amp; 1+a \end{pmatrix} by subtracting the first row multiplied $a$ for each row below the first, and then divides the factor $(1-a)$,  and stuck there. Any hints/helps? In MySQL, how do I get a list of all foreign key constraints pointing to a particular table? a particular column? This is the same thing as , but for MySQL. Does each season of Game of Thrones roughly equal a book? Does each current Game of Thrones season roughly equal one A Song of Ice and Fire book? And do we know which future seasons will cover which books?  I am planning to read the first book after the TV show has finished depicting the events in the first book. I would like to know if after each TV season ends I can expect to be able to read a book covering the events depicted on screen? C++ virtual function from constructor BasicTeX - how to identify packages that need to be added after an upgrade I'm using the BasicTeX distribution on OS X and recently upgrade from the 2011 to the 2012 distribution. This forces me to add the packages that I had added over time to the BasixTeX 2011 installation by pulling them from the online repository with tlmgr. Is there a good way to list these packages that are not part of BasicTeX but are part of the complete TeXLive? How to add percent sign to NSString This is a problem from Introduction to Representation Theory by Etingof et al. I have no idea how to approach it. My friend wrote a program that uses Gaussian elimination to find a basis, which starts with x, y, [x, y], [y, [x, y]], ... but freezes once it gets to elements with 9 or more occurrences of x, y.  I expect that there is some way to characterize the nonzero elements of this Lie algebra (i.e., elements that are "immune" to being shown to be zero through repeated application of the Jacobi identity) which could be turned into a way to enumerate them. I don't see what it is, though.  It's also totally unclear to me how to show that some set we know spans the Lie algebra (proven using some criterion as described in the previous paragraph) is actually linearly independent, unless we find a faithful representation... but I have no idea how to find a faithful representation for an infinite-dimensional Lie algebra.  Hints would be greatly appreciated.  Edit: After looking at the partial output, we can form the "straightforward" conjecture that:   If $n \geq 4$ is even, then there is one basis element of grade $n$, namely $\mathrm{ad}([x, y])^{(n-4)/2}[y, [y, [y, x]]]$. If $n \geq 5$ is odd, then there are two basis elements of grade $n$, namely $[x, z]$ and $[y, z]$ where $z$ is the single basis element of grade $n-1$.   I also have a different conjecture, which is mutually exclusive with the above: that if we flatten out all nonzero elements into the form ${\rm ad}(e_1) {\rm ad}(e_2) \ldots {\rm ad}(e_{n-1}) e_n$ using the Jacobi identity, then in the sequences $e_1, \ldots, e_{n-1}, e_n$ and $e_1, \ldots, e_n, e_{n-1}$, any span containing two x's must contain at least two y's and any span containing five y's must contain at least two x's, otherwise we can use the Jacobi identity to rearrange the element somehow to show that it vanishes. If this is true, then the long-range behaviour must be:   For $n \geq 8$ there is exactly one basis element of grade $n$, namely $[... [y, [x, [y, [y, [x, y]]]]] ...]$ (the pattern $y x y$ repeats).   I still don't see an approach to proving that none of the elements in this sequence actually vanishes, though.  Edit 2: Here is the hint given by Etingof in an email:     No, there are no published solutions, since these problems are used as   a homework for classes. This is indeed a nontrivial problem.      This is the positive part of the affine Kac-Moody algebra of type   A_2^2, which is the twisted loop algebra for sl(2). So what you need   to do is to relate this Lie algebra to loops into sl(2). I see from   your post that you are on the right track.