Math.cos() gives wrong results
Is floating point math broken?
Center figure that is wider than \textwidth I have a figure that is wider than the \textwidth of my document. (I don't want to change the \textwidth.) As a result, the leftmost part of the figure is flush with the left margin, while the rightmost part spills into the right margin too much. I'd like to center the figure, so that it will exceed the \textwidth equally on both sides. Is it true that $aH = bH$ iff $ab^{-1} \in H$ Let $H$ be a subgroup of a group $G$. The I know that for $a,b\in G$ we have $aH = bH$ if and only if $a^{-1}b \in H$.  My question is if it is also true that $aH = bH$ if and only if $ab^{-1} \in H$? Does it matter where the $-1$ goes?  I am guessing that this is not true since we keep putting the $-1$ on the left element, but I am also thinking that it might be true because they look like the same.  If $G$ is abelian, then it doesn't matter, so a possible counter example would have to involve a non-abelian group. If $H$ is a normal subgroup, then it should also be true because then $a^{-1}bH = Ha^{-1}b$. Strange set notation (a set as a power of 2)? I am not very well versed on set theory or syntax, but I thought I knew the basics.  However, in a book about databases I am reading now, the author uses $2^x$ to signify "a set of $x$."  For example, $2^{\text{dogs}}$ is a set of $\text {dogs}$, etc.    The author never really explained this or why he does it, I just picked up the meaning from context.    I am not sure why the exponent operator is used, nor am I sure what the number $2$ has to do with it.  The sets being represented are NOT powers of $2$ (in size)... they come in all sizes.  Is this a valid notation?  I have not seen it anywhere before... Euclidean distance vs Squared Bootable USB of Ubuntu Minimal CD I want to burn Ubuntu Minimal CD ISO to an old, 1 GB USB thumbdrive. Regular ISO doesn't fit. I already tried Startup Disk Creator and Unetbootin but both failed. I tried installing GRUB and pointing it to boot like this:  insmod part_msdos linux /linux initrd /initrd.gz   but it doesn't work. I also tried dd but I ended up with not bootable drive.  Rufus worked on Windows with the same thumbdrive but I no longer use Windows.  What else can I do? The Download CSV link seems broken in SEDE query result window I've created   which works nice. I would really like to download those results as an CSV file. Therefor I click on the provided link on the right hand side of the screen called Download CSV. The result is:  The webpage cannot be found  https://data.stackexchange.com/stackoverflow/csv/202355?UserId=187606 I expected: A file in CSV format. It looks like a regression of  Queries without a parameter work as advertised. Can this be fixed? I'd like to write an N-body simulation in which I collide two disk galaxies. To give you an idea of the accuracy I'm trying to achieve, I'm aiming to make this my screensaver at 30fps on my work desktop. Think Euler's method, large epsilons in denominators to prevent blow up, all that lazy stuff.   One place I don't want to be lazy, however, is in the initial construction of the galaxies I'm trying to collide. I want to design a galaxy that will at least sort of hold together by its own gravity until it smash it into another one.   How might I go about selecting the orbital velocities for each of the bodies composing the galaxy? Allow moderators to see delete/undelete activity on users' profiles We get auto flags for 'possible vandalism' related to deletes, and these can be helpful at times in finding 'rage quit' users, or other potential problems.  But often, I would like to know if there is a prior pattern with the user doing this. While we can see posts of theirs which have been flagged, and perhaps find previous auto flags this way, it would be nice to see an overall view of their self-deletions and undeletions; mostly just to identify patterns.  Such a feature would be nice if it was activity centered, rather than post-centered. That is; each delete/undelete action should be shown. So we could easily see, for example, if someone went back-and-forth on a single post.  The "activity" tab in the profile would be a logical place to show these events.  I think this would be best to be limited to instant self-deletions, not deletion votes. How can an encryption scheme be indistinguishable with multiple messages but vulnerable to an adaptive chosen-plaintext attack? In order to be indistinguishable with respect to multiple messages, wouldn't the scheme need to be nondeterministic? How would this then possibly be vulnerable to an adaptive CPA? What content type to force download of text response? I need to stream a file to the Response for saving on the end user's machine.  The file is plain text, so what content type can I use to prevent the text being displayed in the browser? Proof that $S_3$ isomorphic to $D_3$ Please suggest me any particular unnecessary file that I can clean to back everything to normal condition(temporarily). (i.e. any log or archieve or anything ). My var/log has only 40MB and Home directory has 3GB of space(so I believe that's not a problem). Other than that what I can clean up to make space.   [user@host]$ df -h Filesystem            Size  Used Avail Use% Mounted on /dev/mapper/vg_inamivm-lv_root                        18G   17G     0 100% / tmpfs                 1.9G     0  1.9G   0% /dev/shm /dev/sda1             485M   71M  389M  16% /boot   I am in a debian machine.   UPDATE1:  output of  cd /; du -sxh *  6.1M    bin 61M     boot 156K    dev 22M        etc 3.3G    home 306M    lib 18M     lib64 16K     lost+found 4.0K    media 4.0K    mnt 408K    opt du: cannot access `proc/18605/task/18605/fd/4': No such file or directory du: cannot access `proc/18605/task/18605/fdinfo/4': No such file or directory du: cannot access `proc/18605/fd/4': No such file or directory du: cannot access `proc/18605/fdinfo/4': No such file or directory 0       proc 208K    root 9.7M    sbin 0       selinux 4.0K    srv 0       sys 8.0K    tmp 536M    usr 187M    var   Update2  Output of ls -la /  dr-xr-xr-x.  22 root root  4096 Aug  7 08:42 . dr-xr-xr-x.  22 root root  4096 Aug  7 08:42 .. -rw-r--r--.   1 root root     0 Aug  7 08:42 .autofsck dr-xr-xr-x.   2 root root  4096 Mar 28 16:53 bin dr-xr-xr-x.   5 root root  1024 Mar 28 16:54 boot drwxr-xr-x.  16 root root  3580 Sep  9 03:13 dev drwxr-xr-x.  69 root root  4096 Aug 23 09:19 etc drwxr-xr-x.   9 root root  4096 Jun 29 16:10 home dr-xr-xr-x.   8 root root  4096 Mar  7  2012 lib dr-xr-xr-x.   9 root root 12288 Mar 28 16:53 lib64 drwx------.   2 root root 16384 Mar  7  2012 lost+found drwxr-xr-x.   2 root root  4096 Sep 23  2011 media drwxr-xr-x.   2 root root  4096 Sep 23  2011 mnt drwxr-xr-x.   3 root root  4096 Mar  7  2012 opt dr-xr-xr-x. 355 root root     0 Aug  7 08:42 proc dr-xr-x---.   5 root root  4096 Aug 17 18:27 root dr-xr-xr-x.   2 root root  4096 May  2 09:13 sbin drwxr-xr-x.   7 root root     0 Aug  7 08:42 selinux drwxr-xr-x.   2 root root  4096 Sep 23  2011 srv drwxr-xr-x.  13 root root     0 Aug  7 08:42 sys drwxrwxrwt.   3 root root  4096 Sep 13 03:37 tmp drwxr-xr-x.  13 root root  4096 Mar 28 17:53 usr drwxr-xr-x.  18 root root  4096 Mar  7  2012 var Mouse gestures for Unity desktop We all know that a simple statement such as:  SELECT * FROM stuff;   should not produce an ordered result. However, when I try to demonstrate this, it always comes out in primary key order.  Moreover a statement such as:  SELECT thing,whatever FROM stuff GROUP BY thing,whatever;   always appears to sort thing by the last field in the GROUP BY clause, which is not at all helpful.  The question is, under what circumstances does SQL SERVER order the result even if unasked, and what can I do to stop this?  I am trying to demonstrate to my students that order is indeterminate unless specified, but this is not helping my case.  I admit that I am working with a small set of sample data.  Thanks Good Number Theory books to start with? I'm in Grade 11. I'm interested in elementary number theory and would like properly study it. I'm not intending to enter any competitions. How do I go about porting Ubuntu Touch to different devices? I would like to port Ubuntu Touch to a device that is currently not supported. What are the requirements for doing this and where do I begin? Square bracket notation for dimensions and units: usage and conventions One of the most useful tools in  is the use of square brackets around some physical quantity $q$ to denote its dimension as  $$[q].$$  However, the precise meaning of this symbol varies from source to source; there are a few possible interpretations and few strict guidelines. What conventions are there, who uses them, and when am I obliged to follow them? Why people think dark matter as a massive particle? Couldn't it be gravitational waves? Dark matter is said to be present in the galaxies which helped galaxies to exist in the first place forming the glue to attach all the matter in the galaxies.  Given that dark matter never been seen or detected and only effect of which could be felt why do we believe it would be a particle? We have universe (almost) equally distributed in all directions with dark matter forming the skeleton for all the matter that could be seen. Can the dark matter be a illusion of gravitational waves interfering with each other forming this uniform interference pattern of galaxies/matter that is seen by us?  That would also explain the repulsive dark energy which would be the troughs in the gravitational waves indicating desructive interference in the gravitational waves.  Can we hope to simulating the above to come to the possible conclusion of source(two or many) of these gravitational waves that could give rise to skeleton of dark matter and dark energy which may be present in some higher dimentions? Is the sum of all natural numbers countable? I would like to say something like     The boy, Adam's, favorite toy was a bike.   What is the proper way to say this?