Show that the order of $a\times b$ is equal to $nm$ if gcd(n,m)=1
Order of products of elements in a finite Abelian group
Do I have to move swap partition to the right side? According to  I just expand my sda4 to include unallocated space, my question is after create 2 new partitions sda7, sda8, do I need to move the swap partition to the most right of the disk? (I really don't know this, I saw  which moving the swap partition to the right side)  Thanks. Differential Geometry-Wedge product How can we prove the following relation for differentiating the wedge product of a p-form $\alpha_p$ and a q-form $\beta_q$$$d(\alpha_p\wedge\beta _q)=d\alpha_p\wedge\beta_q+(-1)^{p}\alpha_p\wedge d\beta_q$$ Is it possible to soft reset to a shiny Yveltal/Xerneas/Zygarde? I want to gain some rare and valuable shinies, and I looked towards the games mascots, Yveltal, Xerneas, and Zygarde. I know in previous games (like Diamond and Pearl) you could soft reset the game until the game would give you a Shiny in for the legendary. Can you still do this? How can I prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain?  Also, I need to prove that its field of fractions is isomorphic to the field of rational functions $\mathbb{Q}(t)$?    (The question is taken from ) How can I remove all duplicates so that NONE are left in a data frame? I have this recipe for cheese curds that I want to try: , but it specifically calls for raw milk which is near to impossible to get here. I make yogurt with pasteurized milk and it works no problem, but does cheese somehow work differently? Converting FLV to MP4 Using FFMPEG and Preserving the Quality Im trying to use ffmpeg to convert my flv files to mp4 to play them on iOS devices but the converted video has a much worse quality than the original one.  Here is the command i use:  ffmpeg -i input.flv -ar 22050 output.mp4   I would really appreciate if someone could provide me with the best settings for flv to mp4 conversion. Is stats maths or not?   Given that it's all numbers, mostly taught by maths departments and you get maths credits for it, I wonder whether people just mean it half-jokingly when they say it, like saying it's a minor part of maths, or just applied maths.   I wonder if something like statistics, where you can't build everything on basic axioms can be considered maths. For example, the $p$-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles. Sort a Map by values I am relatively new to Java, and often find that I need to sort a Map&lt;Key, Value&gt; on the values.  Since the values are not unique, I find myself converting the keySet into an array, and sorting that array through array sort with a custom comparator that sorts on the value associated with the key.  Is there an easier way? How do I avoid rendering glitches from overlapping faces? What is the difference between the words which and that?  For example I have the following sentences:   “I have a car which is blue.” “I have a car that is blue.”   Are there any rules specifying usage of which and that? I once heard the late John Ciardi (NPR's "A Word in Your Ear") try to explain that the 1920s idiom, "copacetic" (meaning completely satisfactory), entered into the African-American vocabulary in Harlem from the days when Jews and African Americans lived there together.  He argued that copacetic has the same meaning as the Israeli idiom "kol b'seder" which literally means "all is in order."  The problem with that, said my Harlem-raised father-in-law, is that the Jews in Harlem spoke Yiddish and kol b'seder was not used in Yiddish.  The dictionary I've got is not helpful. Can someone come up with a better explanation? I want to copy the data folders from a previous ubuntu(14.04) partition My hard drive is a mess and I want to reformat it and do a fresh install of 16.04. My problem is I only have access to the HDD via 16.04 on a USB stick and although I can navigate to the various folders, nothing I do allows me to copy to an external hard drive. The folders are not encrypted. I also have a small swap partition - I would like to preserve the data on that too if possible.  Here's a listing of the home directories on the drive:  ubuntu@ubuntu:/$ ls -l /media/*/*/home /media/ubuntu/0e228f65-dac7-435f-828b-1eb1a6c3f29a/home: total 4 drwxr-xr-x 46 1000 1000 4096 Dec 21  2015 simon  /media/ubuntu/db79a6d1-7c6c-4d8c-822b-ba6ccd1ddecd/home: total 4 drwxr-xr-x 27 1000 1000 4096 Apr 17  2014 simon  /media/ubuntu/ebe2a6ba-9cb5-4279-8352-5082b8dd4d5e/home: total 4 drwxr-xr-x 39 1000 1000 4096 Aug  3 14:21 simon Why didn't (or did) I or some other user get a hat although the requirements were (or weren't) fulfilled? How can I use Unicode-aware regular expressions in JavaScript?  For example, there should be something akin to \w that can match any code-point in Letters or Marks category (not just the ASCII ones), and hopefully have filters like [[P*]] for punctuation, etc. What is taking up so much space on my disk, beside the filesystem? I have only one disk in my computer, a 80 GB SSD. It is formatted as a single ext4 partition (no swap), and all the usual folders are installed on it (I keep lots of data on external media, but /home and all the rest is on the SSD).   Today I booted it and I got a message that the drive is full. I opened Disk Usage Analyzer to look at what takes up space. It insists that 67.8 GB of the disk are used, and that / is taking up 36.4 GB. Which leads to the question, where are the missing 30 GB, if not on the file system?!    Just to have a comparison, I added up all the sizes of all readable folders listed in Nautilus as subfolders of / (including the hidden ones). I got 20.9 GB. Trash was unreadable, but I know it has 16.2 GB, so the sum is 36.1 GB, around the same as what Disk Usage Analyzer is reporting. There were some system folders which were unreadable, like proc, but I doubt that they add up to 30 GB - else it wouldn't be possible to install Ubuntu on small disks, and I've seen it run on 2.5 GB. I think they must be the 0.3 GB difference between my calculation and the Disk Usage Analyzer's report.   So I'd like to know, what is eating up these 30 GB and how do I get it to free them?     Edit with answers to CYREX's questions   Booted from Ubuntu 10.10 64-bit live CD (same as my system).  fsck result:   root@ubuntu:~# fsck /dev/sda1 fsck from util-linux-ng 2.17.2 e2fsck 1.41.12 (17-May-2010) /dev/sda1: clean, 265956/4890624 files, 18073343/19537408 blocks  Booted from the SSD. Emptied the trash, and the 16 GB from there are now free. The 30 GB are still missing.    I'm not experienced enough to know what counts as weird in a log. Here are my messages log and my syslog since last boot. Maybe you can find something in them:       Here it got really strange. I plugged in an external 500 GB HDD. Disk Usage Analyzer overreported the available space, then showed space missing (76 GB used, but only 24 in folders).      Booted again from the LiveCD, started the Disk Usage Analyzer from it and got the same results as from the installed Ubuntu, within 1-2 GB.   Edit with answer to CodeMonk:  That would have been a nice solution, but the partition is really 80 GB - I mean 74 GB + marketing "error", anyway the whole disk is formatted. It also shows that over 50 GB are used, so where are the 30 GB if not in files and folders?    GParted also reports the correct size of the external HDD. Let's consider a polynomial ring of single variable. One can define them informally by saying $P(X)=\sum_{i=1}^n a_n X^n$ while $X$ is an indeterminate variable. However, since mathematics is based on first-order logic, one can only talk about something that actually exists (as a set). So as its name says, indeterminate variable is not a sentence in $ZFC$. Nevertheless, we can formally define a polynomial ring $R[X]$ as a subset of $R^\omega$ whose support is finite. ($R$ is a commutative ring with unity) Just like a single variable, I want to know what would be the formal definition of a polynomial ring of several variables. What would be a formal definition? === I have figured out a candidate. That is,  Candidate for a &quot;Definition of a polynomial ring $R[X_1,...,X_n]$&quot;. ($R$ is a commutative ring with unity) Let $R[X_1,...,X_n]$ be the set of all functions $f:\omega^n \rightarrow R$ whose support is finite. Define $(f+g)(a,b,c)=f(a,b,c)+g(a,b,c)$ and $(f•g)(a'',b'',c'')=\sum_{a+a'=a''} f(a,b,c)•g(a',b',c')$ (Note that in this way, $+,•$ are well defined.) Then, it can be checked that $(R[X_1,...,X_n],+,•)$ is a ring.  Is this a formal definition of a polynomial ring? Why isn't smoke flowing with the shape of the domain? I created this Smoke domain.    Then When I added smoke, it didn't stay within the domain, it went through the hole. Proof of Frullani's theorem How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are  Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously differentiable function such that $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = 0, $$ and let $ a,b \in \left( {0,\infty } \right)$. Prove that $$ \int\limits_0^{\infty}  {\frac{{f\left( {ax} \right) - f\left( {bx} \right)}} {x}}dx  = f\left( 0 \right)\left[ {\ln \frac{b} {a}} \right] $$  If you know a more general version please give it to me )= I can´t prove it. I have a DB Server running CentOS 6.4 and MySQL 5.5 from RPMs (not source).  I have dozens of databases but one of them is being hit quite hard with large selects which locks out other updates which in turn locks out other selects (at least I think this is what's happening!)  To alleviate this I would like to replicate one of the databases onto the same server so that the writes can write to the "master" and reads can be done on the "slave". I don't have the luxury of additional hardware.  I have found a lot of guides on the internet describing how to replicate to a different server, and one () describing how to replicate to the same server but it doesn't seem to be geared towards retrofitting this functionality to an existing database with 10s of GB of data in it already. It also talks about compiling MySQL from source.  Does anyone know how I can retrospectively convert my MySQL 5.5 config to replicate one of my databases on the same server?  My existing DB runs on port 3306 so I feel it should be as easy as firing up another instance of MySQL on port 3307 and making some changes to my.cnf but not sure what changes.  Thanks in anticipation.