$\begin{cases}  &amp;x + y + z = 12\\&amp; x^2 + y^2 + z^2 =  12 \\ &amp; x^3 +y^3 + z^3 = 12 \end{cases}$      If $x,$ $y,$ and $z$ satisfy the system of equations above, what is the value of $x^4+y^4+z^4?$    People told me that this can be solved by using Newton's sums or Newton's identities which I don't know how. Does anybody know how to do that or any other method to solve this?
Question : $$\begin{align}a+b+c &amp;= 0\\ a^3 +b^3 +c^3 &amp;= 12\\ a^5 +b^5 +c^5 &amp;= 40\end{align}$$ Then , $a^4 + b^4 + c^4 = ?$  My try : As a common perspective I just went to find any trick related to it and my first step is as usual as a common man will think $3abc = 12$ and got $abc = 4$ After that I am unable to link it with any other equation
The gluing lemma for closed sets states:     Given a finite closed cover $\{A_i\}$ of a topological space $X$, together with continuous maps $\{f_i : A_i \to Y\}$ into some other topological space $Y$, there exists a unique continuous map $f : X \to Y$ whose restriction to each $A_i$ is equal to $f_i$.   Question: What is a good and simple counter-example when the gluing lemma fails in the case that $\{A_i\}$ is infinite, but countably so.  My attempt: I have only been able find a silly counter-example: let $\{A_\alpha\}$ be the collection of all points of $X = [0,1] \subset \mathbb{R}$, and let $f_\alpha = 0$ for all $\alpha$ except for $\alpha = \alpha_0$, for which $A_{\alpha_0} = \{0\}$ and $f_{\alpha_0} = 1$. Custom attributes - Yea or nay? I often end up issuing complex commands over ssh; these commands involve piping to awk or perl one-lines, and as a result contain single quotes and $'s.  I have neither been able to figure out a hard and fast rule to do the quoting properly, nor found a good reference for it.  For instance, consider the following:  # what I'd run locally: CMD='pgrep -fl java | grep -i datanode | awk '{print $1}' # this works with ssh $host "$CMD": CMD='pgrep -fl java | grep -i datanode | awk '"'"'{print $1}'"'"   (Note the extra quotes in the awk statement.)  But how do I get this to work with, e.g. ssh $host "sudo su user -c '$CMD'"?  Is there a general recipe for managing quotes in such scenarios?.. "1 out of 100 chickens is" or "1 out of 100 chickens are"? I'm in an argument. To me "are" makes more sense. I understand the rationale for is because it's only one chicken, but chickens itself is plural. Help? What is the recommended way to empty a SSD? I've just received my new SSD since the old one died. This Intel 320 SSD supports TRIM. For testing purposes, my dealer put Windows on it, but I want to get rid of this and install Kubuntu on it.  It does not have to be a "secure wipe", I just need the empty the disk in the most healthy way. I believe that dd if=/dev/zero of=/dev/sda just fills the blocks with zeroes and thereby taking another write (correct me if I'm wrong).  I've seen the answer , but it looks like it's suited for clearing empty blocks, not wiping the disk.  hdparm seems to be the program to do it, but I'm not sure if it clears the disk OR cleans empty blocks. From its manual page:     --trim-sector-ranges           For  Solid State Drives (SSDs).  EXCEPTIONALLY DANGEROUS. DO NOT           USE THIS OPTION!!  Tells the drive firmware to discard  unneeded           data  sectors,  destroying  any  data that may have been present           within them.  This makes those sectors available  for  immediate           use  by  the firmware's garbage collection mechanism, to improve           scheduling for wear-leveling of the flash  media.   This  option           expects  one  or  more  sector range pairs immediately after the           option: an LBA starting address, a colon, and  a  sector  count,           with no intervening spaces.  EXCEPTIONALLY DANGEROUS. DO NOT USE           THIS OPTION!!              E.g.  hdparm --trim-sector-ranges 1000:4 7894:16 /dev/sdz   How can I make all blocks appear as empty using TRIM? factorization of a^n+1? I'm trying to prove that $a^n+1$ can only be prime if $n$ is a power of $2$. Is there a general factorization of $a^n+1$? AUC and class imbalance in training/test dataset I just start to learn the Area under the ROC curve (AUC). I am told that AUC is not reflected by data imbalance. I think it means that AUC is insensitive to imbalance in test data, rather than imbalance in training data.  In other words, only changing the distribution of positive and negative classes in the test data, the AUC value may not change much. But if we change the distribution in the training data, the AUC value may largely change. The reason is that the classifier cannot be learned well. In this case, we have to use undersampling and oversampling. Am I right? I just want to make sure my understanding on AUC is correct. The durability of iron tools is really starting to annoy me, but in all my hours of playing I've only stumbled across one deposit of diamond ore that provided around 7 Diamonds. What's the best way to locate more in Survival mode? Where is it most commonly found? How to turn cheats on or off I would like to get a USB token to secure my keys. My use case is protection of 3 GnuPG keys that I will be using 10 times per day at least. I plan to create a new key ring from scratch. Because ECC seems more future-oriented than RSA, this is what I chose to use. I'm wondering which USB token to choose as well as which curve.  On  it is said that many people think curves recommended by the NIST or defined by Brainpool have a doubtful origin; therefore, they recommend the non-standardized Bernstein’s Curve 25519. On  one poster says that    He is not aware of any profound criticism of Brainpool curves, and Bernstein’s Curve 25519 is hard to protect against side-channel attacks when implemented in embedded devices.   As a result, I'm a bit lost as to which key/curve to choose. How to find historical airfares? Is there a way to find out how much it would have cost to buy a particular airplane ticket at some point in the past?    For example, how could I find out how much it would have cost to buy a one-way itinerary from airport A to airport B on October 1, 2011 for travel on November 1, 2011? What exactly is the difference between a parametric and non-parametric model? I am confused with the definition of non-parametric model after reading this link  and .   Originally I thought "parametric vs non-parametric" means if we have distribution assumptions on the model (similar to parametric or non-parametric hypothesis testing). But both of the resources claim "parametric vs non-parametric" can be determined by if number of parameters in the model is depending on number of rows in the data matrix.  For kernel density estimation (non-parametric) such a definition can be applied. But under this definition how can a neural network be a non-parametric model, as the number of parameters in the model is depending on the neural network structure and not on the number of rows in the data matrix?  What exactly is the difference between parametric and a non-parametric model? Why is paper more frangible when it is wet? My four-year-old daughter asked me why paper tends to fall apart when wet, and I wasn't sure.  I speculated that the water lubricates the paper fibers so that they can untangle and separate more easily, but I really wasn't sure. I'm looking to view and archive all my old DMs in Twitter, since I would like to revive conversations that took place 1 year ago.  Acording to Twitter they don't delete old messages, they simply archive them in their database.  Is there a way to view and download all of these messages, just like you can view all your old public tweets with third party apps?  UPDATE: I found  but apparently it's not working for DMs now How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$? I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property. How can gravity affect light? How to draw arrows between circled coefficients of polynomials using tikz After reading Gonzalo's answer to , I thought  I would be able to modify it so that I could have the corresponding coefficients circled and an arrow going from one circle to the next. For example, I would like the coefficient of x^2 on the left, a, circled and the coefficient of x^2 on the right, 5 circled, then a curved arrow drawn between the two circles. How might I achieve this? Also, how would I add some text along the path of the arrow? I reproduce his code here.  \documentclass{article} \usepackage{amsmath} \usepackage{tikz} \usetikzlibrary{calc}  \newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};}  \begin{document}  \begin{equation}   a\tikzmark{a}x^2 + bx + c = 5\tikzmark{b}x^2 + bx + c.   \begin{tikzpicture}[overlay,remember picture,out=315,in=225,distance=0.4cm]     \draw[-&gt;,red,shorten &gt;=3pt,shorten &lt;=3pt] (a.center) to (b.center);   \end{tikzpicture} \end{equation}  \end{document} I'm working on a new edition of a book which uses a very unusual style of bibliography: Entries are sorted in author-year fashion, but then numbered per author or group of authors, e.g.:  Milnor, J.:    1. Construction of universal bundles: I, Ann. Math., (2)63:272-284 (1956).    2. Construction of universal bundles: II, Ann. Math., (2)63: 430-436 (1956).    3. On manifolds homeomorphic to the 7-sphere, Ann. Math., (2)64: 399-405       (1956).    4. The theory of characteristic classes, mimeographed, Princeton University,       Princeton, N.J., 1957. Milnor, J., and M. A. Kervaire:    1. Bernoulli numbers, homotopy groups, and a theorem of Rohlin, Proc. Intern.       Congr. Math., 1958. Milnor, J., and E. Spanier:    1. Two remarks on fibre homotopy type, Pacific J. Math., 10: 585-590 (1960).   Citations are formatted as Milnor [3], and so on.  Is there any bibtex style for something strange as this? Or can this type of bibliography be created with biblatex without too much coding? How to understand a volatility based ETF like VXX The ETF VXX is based on the volatility index VIX. If volatility goes up, VIX goes up and VXX goes up. Similar thing for the downside. Now when I buy an ETF like SPY, I can conceptually understand that I am buying a small portion of each of the underlying companies. So if an underlying company stock makes a move, the ETF SPY makes a move in  the same direction.  But when I buy VXX, what exactly is the underlying? It is VIX. I am unable to grasp the concept of 'volatility' as the underlying. I can visualise only stocks and commodities as the underlying because in my understanding, the unerlying needs to be something physical (let us say stocks are physical commodities for this example).  Can someone explain the concept of 'volatility' as the underlying?  Is the word 'underlying' even appropriate in this case or is the word 'underlying' only to be used for derivative products? math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?