A Linear map A whose trace is 0.
If $\mathrm{Tr}(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$
Dollar-cost-averaging interval based on decay of autocorrelation? What command(s) would you use to extract all of the usernames from a database called /etc/passwd and then sort this output alphabetically in reverse order? Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$. Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  My work: So I think I have to do a proof by induction and I just wanted some help editing my proof.  My attempt:  Let $P(n)=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  Then $$P(1)=1^2=\frac{1(1+1)(2+1)}{6}$$ $$1=\frac{6}{6}.$$ So $P(1)$ is true.  Next suppose that $P(k)=1^2+2^2+\cdots+k^2=\frac{k(k+1)(2k+1)}{6}$ for $k \in \mathbb{N}$. Then adding $(k+1)^2$ to both sides of $P(k)$ we obtain the following: $$1^2+2^2+\cdots+k^2+(k+1)^2=\frac{k(k+1)(2k+1)}{6}+(k+1)^2$$ $$=\frac{2k^3+3k^2+k+6(k^2+2k+1)}{6}$$ $$=\frac{2k^3+9k^2+13k+6}{6}$$ $$=\frac{(k^2+3k+2)(2k+3)}{6}$$ $$=\frac{(k+1)(k+2)(2k+3)}{6}$$ $$=\frac{(k+1)((k+1)+1)(2(k+1)+1)}{6}$$ $$=P(k+1).$$ Thus $P(k)$ is true for $k \in \mathbb{N}$. Hence by mathematical induction, $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ is true for $n \in \mathbb{N}$. I am wondering whether there is any difference between a gerund acting as subject and an infinitive acting as a subject. Wacom Tablet - Illustrator brush tool I currently got the intuos art wacom tablet so still learning how to use it. I'm currently in illustrator trying to draw out a map and the brush tool is doing something weird and I cannot figure out why. You can see in my image that where there are curves the black slightly spills over and is filling in half the curve with black. Brush setting is set up as a new calligraphic brush and the size is set to pressure, 3pt-3pt. If someone else is booking international ticket for me using his credit or debit card, does he need to be present physically at the time of check-in?   Suppose I am coming back from Singapore to my home country and my ticket is booked by my friend for me, who lives in my home country. Do airlines in Singapore ask for the person to be present physically to check in? Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence? $$ \sum_{n=1}^\infty \frac{n!e^n}{n^n} $$ Where $e$ is Euler's number.  Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. Our calculus teacher told us that it diverges but he doesn't see any elementary way or simple test to prove it. He told us something about Euler's Gamma function and said that he knows it diverges because of Stirling's formula but this is not quite our level yet.  I tried to think about it myself but I established only that this  is some kind of an edge case. We could generalize the problem to series of $\frac{n!a^n}{n^n}$. Then, from the ratio test we know that it converges for $a &lt;e$ and diverges for $a&gt;e$. For $a=e$ the test is inconclusive. I've also tried to compare it with various divergent series, but nothing worked...  Does anybody have any idea for a simple solution? My friend wants to consolidate his various retirement account accumulated over the past 20-30 years to make it easier to manage/track his retirement savings. He doesn't recall immediately all the accounts he currently has (one of the reasons to rollover), but we've determined he has at least the following types of accounts: 401(a), 401(k), 403(b), 457(b), and "something from the city". He's trying to track down all the relevant paperwork for all of the accounts, but we'd like to know what kind of limits he might have to watch out for.   Are there any restrictions on the types of the accounts that can be rolled over? Is there a limit to the number of accounts that can be rolled into a single IRA (whether all at once or over the lifetime of the IRA)? Is there a limit to the number of times accounts can be rolled over (can/should he do 1 or 2 at a time as he gets the paperwork in order, or wait until he has everything sorted out)? Is this a yearly limit or lifetime? Is there a dollar limit on the amount that can be rolled over? Is there a limit to the age of accounts that can be rolled over?   As far as we know, all of the accounts are pre-tax/non-Roth accounts, and that Roth vs. traditional accounts should be rolled over to separate (Roth vs. traditional) IRAs. Where can I download Windows 8 legally, from Microsoft? Is there any canon evidence that FN-2187 actually worked in sanitation? Finn tells Han that he worked in "Sanitation" on Starkiller.  FINN The flooding tunnels are over that ridge. We'll get in that way. HAN What was your job when you were based here? FINN Sanitation. HAN Sanitation? Then how do you know how to disable the shields? FINN I don't. I'm just here to get Rey   It just occurs to me - the "I was in sanitation" line given to Han seems to be the only actual canon evidence of "Sanitation" job.   But was it the truth"?!?!  Finn is always trying to create a positive impression of himself with the Resistance, from the get-go:   He tells Poe that he's rescuing him because that's the "Right thing to do" (with Poe quickly seeing through the fib: "You need a pilot!") He tells Rey that he's Resistance (she buys that because she's a romantic 19 year old still playing with X-Wing pilot helmet) He tells Han that he's Resistance (that one lasts about a minute - Han was not a romantic 19 year old even at 19 :)   In reality, we know that he was 1% of his class, model stormtrooper cadet, personally groomed for promotion by Captain Phasma (; ).  However, I am pretty sure telling Han "Hey, I was 1% of my class, model stormtrooper cadet, groomed for promotion by Captain Phasma"  isn't high on his list of brand management plans.  Far better to pretend to be a lowly "sanitation" worker than top-flight cadet that he really was.  So, my question is - do we have any direct canon evidence proving that the "sanitation" job was real and not a fib to make himself look better to Han/Resistance?  Knowing where trash compactor was isn't direct evidence, though it is circumstantial one. Difference between $:=$ and $=$ I am sorry if this is quite elementary question. But I always think, that why we use $:=$ at some places, instead of $=$. Is there any fundamental difference between these two? Before reading Terry Tao's blog (4 months ago), I had never seen a symbol like this (:=). Is [Its'] a word? (Note the apostrophe at the end.) I just had a strange flashback to a conversation I had when I was in high school, with a man who was regarded by many members of a particular online community as having an impressive degree of knowledge of the English language.  The conversation centered on a claim this man made that I found very difficult to accept. I had made some remark involving the difference between it's and its (a distinction which I trust is quite well-known to the majority of users on this site), to which he had contributed, mostly phrased as an amusing aside, that there was one more word I hadn't mentioned: its', with the apostrophe at the end.  I originally thought he might have been joking, but we ended up debating this rather fervently. I seem to recall that I kept demanding he use the word in an example sentence, but he either could not or refused to do so. Yet he maintained that it is a word.  Is this true? I must concede I haven't put a lot of thought into it just now; but at the time, I was perplexed by the very suggestion that it could be a word (what could it mean?), and at the moment I can't really think of any scenario where it would make any sense. I'm trying to create an annotated bibliography. Here's my .tex document:  \documentclass[12pt]{article} \title{Annotated Bibliography}  \begin{document}  \nocite{*}  \bibliography{annotatedbib}  \end{document}   And here is the associated .bib document:  @Misc{hacker,     AUTHOR = `{Andrew Hacker'},     TITLE = {Is Algebra Necessary?},     HOWPUBLISHED = {New York Times},     MONTH = {July 31},     YEAR = {2012},     ANNOTE = {This article was published in the New York Times.}, }   No matter into which format I compile my .tex document, the annotation will not show up! Is there something that I'm missing? This doesn't work:  tar xf /tmp/foo.tar.gz foo/bar tar: foo/bar: Not found in archive   It's not obvious to me what would do this beyond extracting it in place and moving the files over. How do I parse an ISO 8601-formatted date? Money from my parents from abroad to my bank account in UK I am from Japan. I have moved to UK recently on a spouse visa.   I still have my Japanese bank account which I have been using even before I came to UK. I have saving in that account and also my parents transfer me some money per month in my bank account in Japan. I have an International Debit Card on this account.  I am withdrawing this Money in UK via ATM / Debit Card.  In this case, do I have to pay taxes in UK or declare this as income in some form. Prove that the union of two disjoint countable sets is countable 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. With jQuery, how do I find out which key was pressed when I bind to the keypress event?  $('#searchbox input').bind('keypress', function(e) {});   I want to trigger a submit when ENTER is pressed.  [Update]  Even though I found the (or better: one) answer myself, there seems to be some room for variation ;)  Is there a difference between keyCode and which - especially if I'm just looking for ENTER, which will never be a unicode key?  Do some browsers provide one property and others provide the other one? I develop an application in which i constantly get samples of heart pulse.  I defined an interval of t seconds.  In each t seconds I have n samples.  In every interval, I want to calculate the tendency of those n samples. For example, lets say I have n = 5, and I have samples with values {70, 88, 95, 103, 115}. I want to recognize I have a growth in the heart pulse, and I want to have some measure for the rate of grows/decrease/almost no change.  I thought on two approaches for solving this problem.   I calculate the linear approximation for the n samples using linear regression by applying least squares implemented by the normal equations. (I treat each sample as two coordinates vector with x coordinate as time and y coordinate as heart pulse). I get from the normal equations a linear function of the form y = mx+b and my measure for the tendency is the slope, i.e. the m value. The second approach is to calculate the correlation (pearson's correlation) between vector x and vector y when x is the time and y is the heart pulse.   I'm asking which approach do you think is better for my problem (determining the tendency of heart pulses). Or, do you have a better algorithm for solving this problem?