In Pgfplots, how can I prevent points from hiding lines
PGFPlots: Understanding how to draw plots on different layers and arrange them
Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)&lt;3$   for all positive integers $n$   This problem is copied from Math Olympiad Treasures by Titu Andreescu and Bogdan Enescu.They start by stating that induction wouldn't directly work here since the right hand side stays constant while the left increases.They get rid of this problem by strengthening the hypothesis.$$\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)\le3-\dfrac{1}{n}$$ and then proceed by induction.  The problem is that I can't find a motivation for the above change.I mean,we could have subtracted a lot of things from the RHS but what should nudge us to try $\dfrac{1}{n}$?The rest of the proof is quite standard,but I can't see how I am supposed to have thought of it.Is it just experience?Or is it a standard technique?A little guidance and motivation will be appreciated. Change running heads in longtable (or other "long table" package) Prove that matrix can be square of matrix with real entries Prove that matrix   \begin{bmatrix}1&amp;0&amp;0\\0&amp;-1&amp;0\\0&amp;0&amp;-1\end{bmatrix}  can be square of matrix with all real entries.  I have found one such matrix to be   \begin{bmatrix}1&amp;0&amp;0\\0&amp;1&amp;-1\\0&amp;2&amp;-1\end{bmatrix} but is there an elegant way to do it without any trial and error? Is it possible to only allow specific packages updates from a PPA How do I prepare best for my first wedding photography event? I am going to shoot a wedding this weekend, and I want to do my best for my friend's big day.    What can I do to prepare for the day in order to minimize failures and mistakes? How do I avoid missing important shots? What backup plan should I have?   I have been shooting for 6 years or so, and I know wedding is one of the hardest thing to do. I have told my friend what I end up producing will be of no match with the work of dedicated wedding photographers, not even the ones that are about to fall out of business.  They are on a tight budget, they said some photos are still better than none, they really must minimize the cost as much as they can. I exaggerated again that my photos will be worse off than the cheapest wedding photographer you can find.  She has been a close friends for years, I want to do everything I can to help her.  Their wedding is at 12:30pm coming Sunday, I have visited the venue last Sunday at 12:30pm to check traffic condition and also the lighting.  The setup is mainly outdoor (no church), the indoor area will not have a good ceiling for bounce. Being at the middle of the day, this can mean some tricky fill flash. I have jot down a list of must-have shots. I am also asking for the seating plan and the event run-down.  Gear:   Canon EOS 60D , EF-S 15-85mm f/3.5-5.6 IS USM Canon EOS 450D (or I can borrow a 550D), EF 50mm f/1.4 Speedlite 580 EXII Speedlite 270 EXII Yongnuo 460 (a cheap but powerful manual flash as backup) Samsung EX-1 f/1.8 point and shoot (backup of a backup...?)   Other lenses that I have but I don't think will be useful:   50mm f/1.8 EF-S 50-250mm f/4.0-5.6 IS II EF-S 18-135mm f/3.5-5.6 IS (I have the better 15-85 on 60D, should I put this on the 450D as a backup allrounder?) EF-S 60mm f/2.8 Macro (should I replace 50mm f/1.4 with this for added macro? You can hardly use f/1.4 under direct sunlight if you want to have fill-flash)   I have two batteries for all cameras (60D 450D and EX1). I have like in total 64GB SD card (I am really inclined to get more, or bring a laptop for on-the-fly backup, or do both). I recently repaired my 580EXII, it has a brand new flash head so the recycle time is as good as it can be, I just need a tonne of batteries. No I don't have external power pack or battery grip for my cameras.  I have a great monopod, but a cheap tripod.  I am happy to bring everything, just putting them in the boot of my car. The carpark is on-site, walking from venue to my car is under 1 minute.  I will have my girlfriend carry the 450D and shoot details / non-portraits.  One concern that I have is fill flash, I really do not have a large surface to bounce off  from. I have shot events before but they are not as dynamic/critical as a wedding.  How do I prepare for differing weather conditions?  If its sunny, fill flash is probably going to be in every single shot. If its cloudy, I probably will turn off the flash. If it rains, my 60D should be fine I have used it in slight rain before, but I really don't know their plan and if they have extendable shelters etc.  What else should I prepare for? Having no experience I want to be as prepared as I can be. Maybe I have overlooked something, maybe there is something else I can do to ensure the shooting will go smoothly. Your help is greatly appreciated! root@SERVER ~$ df Filesystem  512-blocks  Free    %Used   Iused   %Iused  Mounted on /dev/YXCV   655360      365632  45% 6322    13% / /dev/ASDF   3801088     670648  83% 41759   32% /usr /dev/ASR    1048576     500496  53% 5555    9%  /var   How can I pipe the df command's output to only display lines that have more usage than 80%?  e.g.: it would only display:  /dev/ASDF   3801088     670648  83% 41759   32% /usr Correlations between continuous and categorical (nominal) variables home socket with DC voltage I'm wondering why, in a standard house, instead of using many low power AC-DC rectifier, there isn't a big (like 12V 20~50A) AC-DC rectifier which supplies some wall socket with a DC voltage? In the long run this should be less expensive, given the fact that AC to DC in one big apparatus is done better than in many low-quality transformers, right?  Also, modern chip DC voltage regulators have efficiency of >90%, so smaller (or even larger) voltages are still efficient and possible.  So we could use a 12V socket to charge a phone, desktop computer, laptop, radio etc... and still have an AC socket to move washing machine engine and other things that work better with an AC supply.  Why isn't this done?  edit: real question is: is it convenient, given a standard family (2 laptop, 1 desktop, 4 smarthpone, 1 big tv and some minor electonic) to do so? I was looking for alternate proofs of the theorem that "a vector space $V$ of dimension greater than $1$ over an infinite field $\mathbf{F}$ is not a union of fewer than $|\mathbf{F}|$ proper subspaces" and possible generalizations.  A simple measure-theoretic proof over $\mathbb{R}$ is as follows: By countable additivity the sum of the measures of any collection of subspaces is zero since the measure of each subspace is zero, which is a contradiction.  I would like to look at proofs over arbitrary infinite fields and would like to know if similar statements hold for say modules (finitely generated or otherwise) over infinite rings. $\mathbb Z[i]/ \langle 1+2i \rangle \cong \mathbb Z_5$ I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to  $\mathbb  Z_5$.   The only thing that came to my mind was trying to apply the first isomorphism theorem using an appropiate function. If I consider the euclidean function $N: Z[i] \setminus \{0 \}  \to  \mathbb N$ defined as $N(a+bi)=a^2+b^2$, then I can express any element $z$ in $\mathbb Z[i]$ as $z=(1+2i)q+r$ with $N(r)=1,2,3,4$ or $r=0$.   If I define $f:\mathbb Z[i] \to \mathbb  Z_5$ as $f(x=(1+2i)q_x+r_x)=\overline{r_x}$, then it is clear that $f(x)=0$ if and only if $x \in  \langle 1+2i \rangle$. The problem with this function is that it doesn't satisfy $f(x+y)=f(x)+f(y)$ and it is not surjective for if $r_x=f(x)=3 (5)$, then if $r_x=a+bi$, we have $a^2+b^2=3$, which is absurd.  I don't know what else to do, any suggestions would be appreciated. What is the easiest way to swap occurrences of two strings in Vim? How to delete file(s) in secure manner? I deleted a folder in Ubuntu, how can I recover it? The folder is massive (390GB) and contains many different files. The file system is ext4  I was about to copy this file to another hard drive using gksudo nautilus so I didnt have do a massive chmod operation on the destination hard drive, but then I accidentally hit move to trash, which in gksudo nautilus means permanent deletion. I checked the trash but it's not even an operation...  I have seen tools like testdisk and other recovery tools but none that can recover and whole folder because its made up of different files.   I don't think that this would work but if I deleted the partition could I just recover the whole partition at once?  Or would this be wrong because i have already deleted the partition's content? This is my first job in Linux IT.  I don't have a backup... I was trying to make one because the old backup drive died :( Can a homomorphic encryption scheme be made CCA2 Secure? Is it possible to modify a homomorphic encryption scheme so that it can be CCA2 secure?  From the definition of a homomorphic scheme, it seems that it is malleable, which would result in lack of CCA2 security. So, if we use the regular method to convert a CPA secure scheme to CCA2 secure scheme (using a hash of the message and a random value), will it still preserve its homomorphic property? Please correct me if my understanding is wrong. I'm using Ubuntu 16.04 and there's no fglrx! What happened? Apologies for this possibly too vague question, but are there some general guidelines to follow when rendering artificial 3D meshes together with an imported 2D image as the background?  In the attached Blender file (or rendered image), the cube looks so out-of-the-scene. Some reflection from the green background onto the cube should make the cube more realistic in the scene? I don't know how to put it in professional terms -- any suggestions are welcomed! When does $\,n\mid q^3 \Rightarrow n\mid q\,$? I think this is true because of prime factorisations, i.e.  If $3$ a factor of the prime factorisation of $q^3$, then $3$ is a factor of the prime factorisation of $q$.  Therefore If $3$ divides $q^3$, then $3$ divides $q$.  Or for a general case, if $n$ divides $q^3$; then $n$ divides $q$. Is this correct? fp - How to keep running sum in variable? I have this code in my LaTeX:   \FPset{totalHours}{0}  \newcommand{\entry}[5]{     \FPeval{totalHours}{(totalHours)+#5}     #1 &amp; #2 &amp; #3 &amp; #4 &amp; #5 \\ }   Where \entry is meant to be an entry in a time log. The 5th argument is a number representing the number of hours in that entry. I want to keep a running total of the hours in a variable I can print later on, totalHours.   This seems to do the \FPeval but then totalHours gets reset to 0 by the next time I write \entry. What is the correct way to keep a running total? Thanks. I finally got fed up when wanting to read about bash's read and it's -s option with man bash. I found the right spot eventually (around line 4500), but it was a frustrating as usual, since both /read and even /\s-s\s searches have way too many matches.  So, the question is: How can I read long man pages efficiently, or get same information in other ways, locally? As a specific example, how to reach the relevant documentation after seeing read -s pwd in a shell script? A good answer could be a shell script snippet, or hint about some tool and how it is used, or something else entirely, as long as it helps in finding the right spot to read.  Note: I'm not tagging with  because I want the question to be about man page reading in general, even though that quite possibly is the most commonly encountered humongous man page. I changed permissions of a file (chmod g+w testfile) and running ls -l testfile gives:  -rwxrwxr-x 1 user1 user1 0 2011-01-24 20:36 testfile   I then added a user to that group ("/etc/group" has user1:x:1000:user2 line), but am failing to edit that file as user2. Why is this so?