Give two integers $m$ and $n$ such that $n^2$ is a multiple of $m$, $n$ is not a multiple of $m$ and $n &gt; m.$   I think that such two integers don't exist, but even if that were true, I'm having trouble with proving that. I have found clear examples when I neglect the last condition ($n &gt; m$).  Any help on this problem would be much appreciated. The observation that I did make is the fact that since $n^2$ is a multiple of $m$ we can write it as $k * m$, which means that $k$ has to be larger than $n$.  I think that could be a starting point of some sort, but I wasn't capable of going any further.
I am looking to prove the following statement false:  Let $k$ be a positive integer, then $k\mid n$ if and only if $k\mid n^2$.  So I am trying to find a $k$ where this does not hold but after aimlessly trying numbers I was curious if there is a methodical way to find $k$.
Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$  Now, here I got stuck. How to prove it?? Please help. How are answers with the same vote score sorted? Are both try to save the file and try saving the file grammatically correct? If so, is there any difference in meaning? Comparing Arrays of Objects in JavaScript homeomorphism of cantor set extends to the plane? Suppose C is a Cantor set in the Euclidean plane, or even in R^3. Suppose h is a homeomorphism of C onto itself. Can h be extended to a homeomorphism of the whole space? What about if h preserves the ´end points´ of the Cantor set? Seems that the latter is a necessary condition but I don´t know how to prove it. Is it possible to use multiple desktop environments on same system? I am using Ubuntu 12.04. I want to try out KDE and other desktop environments.   But I'm not sure if I'll like them enough to switch permanently. I want to be safe and make a transition in a piecemeal fashion.  Is it possible to install KDE on Ubuntu and yet not do away with the default Unity shell?  Could I install multiple desktop environments and switch between them by tweaking some startup configuration and then rebooting? Does COVID-19 have a case fatality rate of 41%? Is this formula correct? According to the data on the , as of 3rd February 2020 there were 17491 confirmed cases of COVID-19 globally, 536 total recoveries and 362 deaths. From my non-expert calculation this implies a mortality rate of:  (Nd / (Nd + Nr)) * 100 = 41%  where:  Nd is the total number of deaths, Nr is the total number of full recoveries.  This leaves 16593 people still suffering from the disease who have neither recovered or died.  This is in stark contrast to the publicly disseminated value of ~2% mortality, so have I made a mistake in my calculation or assumptions, or is COVID-19 much more dangerous than commonly claimed?    [After a helpful discussion in the comments, 'mortality rate' is not the correct term to use here, instead I should say ''.] Why is ROC AUC equivalent to the probability that two randomly-selected samples are correctly ranked? I found there are two ways to understand what AUC stands for but I couldn't get why these two interpretations are equivalent mathematically.   In the first interpretation, AUC is the area under the ROC curve. Picking points from 0 to 1 as threshold and calculate sensitivity and specificity accordingly. When we plot them against each other, we get ROC curve.  The second one is that the AUC of a classifier is equal to the probability that the classifier will rank a randomly chosen positive example higher than a randomly chosen negative example, i.e. P(score(x+)>score(x−)).  (from ) I use OS X Mavericks and have some company storage drives mounted (using smb or cfs). (smb://username@address.path.toserver.com/directory/data). When I work with files on the drive or copy files to the drive, some hidden files are created. Most of these files start with dot-dash ._ and then the filename of the original file, like:  ._test001.png ._test002.png ._test003.png   And most of the times, the famous ._.DS_Store and .DS_Store are also there.  On my Mac, these . dot items are invisible, but my windows colleagues do see these files. How can I prevent OS X from making these strange invisible files?  On  (and other internet sources) you can find a 'defaults write' to disable the creation of .DS_Store files on your network drives. But this does not work in Mavericks:  defaults write com.apple.desktopservices DSDontWriteNetworkStores -bool true   For now I use this oneliner, but doing a rm -rf does feel wrong in this case.  find . -type f -name "._*" -exec rm "{}" \;   Edit: I don't want to delete the files or exclude them from copying. I want them not to be created in the first place.  Edit: I prefer a 'native' OS X solution over third party apps.  Edit: This questions has two answers, one for the ._ files and one for the .DS_Store files. The bounty is rewarded for dot_clean, this native command line app cleans all ._ files. Unfortunately is does not clear the .DS_Store files, this can effectively be done with Asepsis. Asepsis stores .DS_Stores in a fixed location, and not in the folder itself. Recovering a password for a Word 2007 document In our application, we receive text files (.txt, .csv, etc.) from diverse sources. When reading, these files sometimes contain garbage, because the files where created in a different/unknown codepage.  Is there a way to (automatically) detect the codepage of a text file?   The detectEncodingFromByteOrderMarks, on the StreamReader constructor, works for UTF8  and other unicode marked files, but I'm looking for a way to detect code pages, like ibm850, windows1252.     Thanks for your answers, this is what I've done.  The files we receive are from end-users, they do not have a clue about codepages. The receivers are also end-users, by now this is what they know about codepages: Codepages exist, and are annoying.  Solution:    Open the received file in Notepad, look at a garbled piece of text. If somebody is called François or something, with your human intelligence you can guess this. I've created a small app that the user can use to open the file with, and enter a text that user knows it will appear in the file, when the correct codepage is used.  Loop through all codepages, and display the ones that give a solution with the user provided text.  If more as one codepage pops up, ask the user to specify more text. Is ignoring emails acceptable in academia? As a grad student, I just find it tremendously frustrating when emails are ignored by faculty (both at my institution and at other places). Is this acceptable? I'd be content with a simple response with one line along the lines of "I have read this email and am busy right now, will respond in X days when I can write a more detailed email." I understand that, as a first year student in my field, I'm not really in a position to request anything of anyone, but it just seems like common decency.  Instead, I never hear back, as if my email has been jettisoned off into oblivion. I'm a new blender user, but have used other modelling programs before.   I've downloaded a  and unzipped it.   When I import the obj into blender, it is my understanding (from here: ) that the mtl should just import itself with it. However, when I import the obj, I just get a grey model with no texture. Is it actually importing the mtl but I'm not seeing it? Also for the record, I have changed no settings in blender.  Picture for reference: What would be your preferred way to concatenate strings from a sequence such that between every two consecutive pairs a comma is added. That is, how do you map, for instance, ['a', 'b', 'c'] to 'a,b,c'? (The cases ['s'] and [] should be mapped to 's' and '', respectively.)  I usually end up using something like ''.join(map(lambda x: x+',',l))[:-1], but also feeling somewhat unsatisfied. Let the ring of fractions be denoted as $S^{-1}R$.       Proposition (from Robert Ash's textbook "Basic Abstract Algebra")      Define $f: R \rightarrow S^{-1}R$ by $f(a) = a/1$. Then f is a ring homomorphism. If S has no zero divisors then f is a monomorphism, and we say that R can be embedded in $S^{-1}R$. In particular:      (i) A commutative ring R cna be embedded in its complete (or *full) ring of fractions ($S^{-1}R$, where S consists of all non-divisors of zero in R).       (ii) An integral domain can be embedded in its quotient field.    Why does the proposition say "If S has no zero divisors..."? If S really had zero divisors, then this wouldn't even be properly defined, right? Because we would have a/0 for some a in R. the relationship between maximizing the likelihood and minimizing the cross-entropy I am flying from New Orleans to Dulles (IAD) on United Express, and from IAD to Frankfurt on United. I have a short connection (45 minutes) in IAD.   I'm trying to figure out how can I make it:  Will I need to go throughout security and border check since I'm leaving the US? How big is IAD airport? and are there different terminals for domestic and international flights? If so, how long would it take to walk between them? Cartoon in a world with magic stones wielded by four children which combine to become stronger, and can summon creatures Can you please help me find this animated show? There is a four member team with two boys and two girls. The boys can summon, with their stones, a rock golem (black wearing summoner) and a robot (orange wearing summoner). The two girls can summon something like a dragonfly (green wearing summoner, little girl) and the blue girl, I don't remember what she can summon.  I don't know where the animated show comes from.  By throwing the stone, they activate the stone's power. Each stone has different characters on it. When you don't match them correctly, you get a rebound effect. View linked Excel 'Windows' on separate screens Sort Array Elements (string with numbers), natural sort