x^4 + y^4 + z^4 + 1 = 4xyz No. Of triplets of real numbers satisfying the eqn? I'm not able to figure how to approach this problem.
QUESTION Find the number of triples $$(x,y,z)$$ of real numbers satisfying the equation $$x^4+y^4+z^4+1=4xyz$$  I have tried to solve this problem but can't quite understand how to manipulate the given data to reach a clear result. Could someone please explain how I should approach and solve this problem.Thanks :)
How can Innodb ibdata1 file grows by 5X even with innodb_file_per_table set? I have innodb_file_per_table set and just today my ibdata1 file jumped from 59M to 323M after I made several changes to an 800M table to reduce it to about 600M. That particular table's .ibd file was reduced but the server's ibdata1 file went crazy. Any ideas? What is "preimage resistance", and how can the lack thereof be exploited? How is this different from collision resistance, and are there any known preimage attacks that would be considered feasible? Is it possible to tell Google's Webmaster Tools that our site hosted on www. has moved to a sub folder, sub domain or a new domain with sub folder? Some ships in Star Trek are referred to with definite articles. "The Enterprise", "The Defiant". Other ships are referred to without them. "Voyager", "Enterprise". Is there a pattern to or explanation for this behavior? How are long-term capital gains taxed if the gain pushes income into a new tax bracket? How can I easily clone my entire system? I am running an Ubuntu 14.04 LTS and I experienced some issues with my hard drive. I bought a new HDD of the same size (1TB), and I intend to move all my data (softs, work, pics, etc.) on the new HDD.  What is the easiest way to do that? I'm Australian and have obtained my Canadian working visa. However rather than flying direct to Canada I am meeting an American friend in LA and driving across the border. I have a return flight booked back to Australia from LA in March of next year too. Will the esta visa will be enough to enter the USA because I have my Canadian working visa and plan to travel there? Or will there be a problem because my return flight is booked later then the 90 days allowed on the visa. Resources for Blender I want to resolve merge conflicts in my Git repository. How to do that ? I set up a micro instance (Ubuntu, 32 bit) and assigned an elastic IP (23.23.245.104)  Public DNS: ec2-23-23-245-104.compute-1.amazonaws.com  I successfully ssh'ed to the machine and installed nginx.  Then I edited the server_name attribute in nginx's config file (/etc/nginx/sites-available/default)  server_name ec2-23-23-245-104.compute-1.amazonaws.com;   Restarted nginx and hit  but got a "could not connect" error.  What did I forget to do?  Thanks I recently received an old iMac secondhand.  I have no way to reach the original owner. I am trying to upgrade the OS software, but the prompts demand the administrator's name (which is auto-filled) and password (which I do not have).  I understand why these security features are in place, but is there any way in the world for me to access Admin. privileges without the password? Probability of winning the game "1-2-3" In D&amp;D 4E as with many RPGs, characters have a fixed armour value or AC. When attacking, the attacker must roll a die to overcome the target's AC. In 4E offensive magic works the same way: the target has a fixed Fortitude/Reflex/Will defence determined by their stats, and in order to hit, the attacker rolls to overcome whichever defence is relevant.   However it was not always like this. Some spells in 3E and now again in 5E work the opposite way. In those editions the target is the one who rolls against a fixed value determined by the caster's stats, in order to avoid spell-damage.  Both methods compare the attacker and target's stats, and have an element of randomness. I personally like the 4E system as it simplifies combat without leaving anything out. Why was the decision made, when designing the early D&amp;D editions (and still when iterating upwards into (A)D&amp;D 1e&ndash;3e), to use saving throws to control spells?  Answers should consist of designer quotes and not personal opinions. 1.  Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.  2.   A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)   3.   One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$). How to download and use any other kernel than the one which is installed by default UK Train - Can I board at a different station Prove that $J^+AJ$ has positive eigenvalues Let $J \in \mathbb{R}$ be a $m \times n$ matrix of rank $n$ and a $A \in \mathbb{R}$ be a $m \times m$ diagonal positive definite matrix. Denote by $J^+$ a  of $J$.   As was shown  $J^+AJ$ is not positive definite in general.   However, I was wondering whether one can prove or disprove that $J^+AJ$ has real and positive eigenvalues. Sorting seach results in Android app The issue is how can user sort questions for a particular search query?   For example, I would want to search questions for  and want to see newest questions first.  Right now If we write any query in Search Questions... input box then users are not able to select sorting options as search questions input box takes whole width of the screen and sorting options box gets hidden.  I have also tried to select sorting options first and then search for  but still its showing questions without considering sorting option I have selected. What is the most efficient way to add rows and columns to a matrix? I'm learning C#, so I made a little C# program that says Hello, World!, then compiled it with mono-csc and ran it with mono:  $ mono-csc Hello.cs $ mono Hello.exe Hello, World!   I noticed that when I hit TAB in bash, Hello.exe was marked executable. Indeed, it runs by just a shell loading the filename!   Hello.exe is not an ELF file with a funny file extension:  $ readelf -a Hello.exe readelf: Error: Not an ELF file - it has the wrong magic bytes at the start $ xxd Hello.exe | head -n1 00000000: 4d5a 9000 0300 0000 0400 0000 ffff 0000  MZ..............   MZ means it's a Microsoft Windows statically linked executable. Drop it onto a Windows box, and it will (should) run.  I have wine installed, but wine, being a compatibility layer for Windows apps, takes about 5x as long to run Hello.exe as mono and executing it directly do, so it's not wine that runs it.  I'm assuming there's some mono kernel module installed with mono that intercepts the exec syscall/s, or catches binaries that begin with 4D 5A, but lsmod | grep mono and friends return an error.   What's going on here, and how does the kernel know that this executable is special?    Just for proof it's not my shell working magic, I used  (aka sh) to run it and it still runs natively.    Here's the program in full, since a commenter was curious:  using System;  class Hello {   /// &lt;summary&gt;   ///   The main entry point for the application   /// &lt;/summary&gt;   [STAThread]   public static void Main(string[] args) {     System.Console.Write("Hello, World!\n");   } }