Given that $$2017 = 9^2 + 44^2,$$ use this relation to find at least one group of positive integers $m$ and $n$ that satisfy $$2017^2 = m^2 + n^2.$$
For any natural number $n\ge 1$, given pairs $(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$ of integer numbers, there exist integer number $c$ and $d$ such that  $$\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$$    My initial approach is  Base Case: $(a_1^2+b_1^2) = a_1^2+b_1^2$ which is true. (Although it is trivial)  Prove the statement is true when $n=2$: We have $$(a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2$$ (Thanks André Nicolas for pointing it out)   So if $a,b,c,d$ are integers, $ac,bd,ad,bc$ are all integers and integers are closed under addition and subtraction. Hence $(ac-bd),(ad+bc)$ are integers.   Inductive Hypothesis: $\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$ is true  Inductive Step: $$\prod_{i=1}^{n+1}(a_i^2+b_i^2) = \prod_{i=1}^{n}(a_i^2+b_i^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2)$$  Where $c$ and $d$ are integers.   But when we apply $n=2$, we have  $(c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (e^2 + f^2)$ where $e$ and $f$ are integers.  Hence, by the principle of induction, the statement we needed to prove is true.
I have an external hard disk, and for some reason, it is reporting wrong free space.  In Windows Explorer, it is showing the used space is 378 GB.    I tried running chkdsk H: /f /r, it seems to report the same usage as Windows Explorer, and it says that there is no problem.    But when I tried selecting all the files inside this drive, it is showing much lesser usage.    I've already set the setting to show all hidden files and folders, also show the protected OS files, but it still cannot find about those missing files. And the difference is huge (282 GB). I've already cleaned the Recycle bin as well.    What I am missing here and how do I identify those missing spaces?  [UPDATE]: I used  to help identify and it seems to find some gigantic files, of which filename is quite weird. Anyone knows what this is and is it safe to delete? To be specific, I was trying this code:  package hello;  public class Hello {      Clock clock = new Clock();      public static void main(String args[]) {         clock.sayTime();     } }   But it gave the error      Cannot access non-static field in static method main   So I changed the declaration of clock to this:  static Clock clock = new Clock();   And it worked. What does it mean to put that keyword before the declaration? What exactly will it do and/or restrict in terms of what can be done to that object? Which is correct: "one or more is" or "one or more are"? Should the phrase be "one or more is...", or "one or more are..."? meaning of powers on trig functions I always forget this, when a trig function has an exponent does that mean multiply itself or apply itself to the result recursivly?  e.g. does $\sin(x)^2=\sin(x)\sin(x)$ or $=\sin(\sin(x))$? What about $\sin^2x$? Masking problems I am working with masking. Here is my node setting with the mask.   And here is my node settings  And what am i getting is this.    I tried invert node and many settings. It just doesnt work. Where am i going wrong Proof that a formula $\theta$ is valid if and only if $\forall x \theta$ is valid I need to prove for first-order logic that $\vDash\theta$ if and only if $\vDash\forall x \theta$. The proof for the forward direction seems simple enough:  Suppose $\theta$ is valid. Let $\mathfrak{A}$ be an arbitrary structure and let $s:V\to |\mathfrak{A}|$ be an arbitrary function assigning values of the universe of $\mathfrak{A}$ to variables. Then we know by assumption that $\vDash_\mathfrak{A}\theta[s]$.  But since $s$ is arbitrary, it follows that $\vDash_\mathfrak{A}\theta[s(x|c)]$, for all $c\in|\mathfrak{A}$. But this means that $\vDash_\mathfrak{A}\forall x\theta[s]$. Since $\mathfrak{A}$ and the function assigning elements of the structure to variables are arbitrary, it follows that $\vDash\forall x\theta$.   Now we have to show that if $\vDash\forall x\theta$, then $\vDash\theta$, and this is where I'm stuck. I understand why it makes sense that this should be true, I'm just having a hard time writing it rigorously. Any suggestions would be greatly appreciated. Applying Arzela-Ascoli to show pointwise convergence on $\mathbb{R}$. The statement I am trying to prove:  Let $\{ f_n \}$ be a sequence of equicontinuous, real valued, uniformly bounded   continuous functions on $\mathbb{R}$. Show that $\{ f_n \}$ has a convergent subsequence   which converges uniformly on any bounded subset of $\mathbb{R}$ and pointwise on all   of $\mathbb{R}$ to a continuous function. (Royden pg 210, problem 9).  I believe the first portion follows directly by Arzela Ascoli, once it is noted that if we are given some bounded subset $E$, we can look at $\overline{E}$ a compact set on which the family is equicontinuous and uniformly bounded.  For pointwise convergence on all of $\mathbb{R}$, the question seems to require some sort of diagonalization argument: Take some dense sequence of $\{x_i\}$ (the reals are separable) and write columns of $f_n$ applied to each individual point. There will be a convergent subsequence by Bolzano-Weirstrass and since the outputs are real and bounded by the uniform boundedness of the family. I am a bit fuzzy on this part: Repeating this literally pointwise and moving far enough diagonally, we can correct for different convergence "speeds" pluck out a single convergent subsequence for any given epsilon and have $\sup|f_n(x)-f(x)|&lt;\epsilon$ for any given epsilon. Finally, continuity should follow from an $\epsilon/3$ argument and equicontinuity of the family.  However, in formalizing the above, I am getting caught up in the notation of that sort of argument and am hoping for a more elegant solution. Is there a better way? Am I just generally confused and that's why I'm having trouble writing things down formally?   Thanks! Going into a directory linked by a link When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds?  Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional manifold? How can I make two objects interact I am making an animation where a person sits down on sofa. I need to make it such that when the person sits on the sofa or presses it with his hand, it deforms. However in my case, when he presses on the sofa, his hands goes trough the sofa. How can I fix this problem? How do you take off body armor in GTA Online? It's kind of hard to shop for clothes with a big bullet proof vest on.  Is there a way to take off the armor after missions, or at least hide it? How about a "Vote not to close" option to counter the "Vote to close"? My request is simple: Many times, there will be a question where some people feel like it should be closed. At the same time, I'll feel like it's a perfectly legit question, and it should therefore NOT be closed. Right now, I have to wait till it gets enough close votes (which it always does for some reason, as soon as 1 or 2 people vote to close, e/o else decides to vote as well) and then vote to re-open. How about, as soon as there's one vote to close, there should be an option to counter the close vote, something like "vote not to close" or whatever. This would then bring the "Vote to close" number down by one. Only when the vote to close outnumbers the vote not to close by 5, that's when a question is indeed closed. Any thoughts? I have got this problem on limits  $$ \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$$ What I am doing here is that taking log and then applying L-Hospital's Rule $$y= \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2} $$ then $$\ln(y)=\lim_{x \rightarrow 0} \frac{1}{x^2}\ln\left(\frac{\sin(x)}{x}\right)$$ now it becomes $\frac{0}{0}$ type , but problem is how to take derivative here? when differentiating $\ln\left(\frac{\sin(x)}{x}\right)$ it will become $\frac{x}{\sin(x)}$ then we'll have to differentiate one more time like we do in case of $f(f(x))$ ? $\sin(x)$ and $x$ will be separately differentiated or together ? How do I convert a SMP server world into a Single-player compatable world? I had a Minecraft world on Minefold.  I archived my Minecraft world just before Minefold closed about a year ago.  I'm trying now to open the world in my local Minecraft installation on OS X, but it doesn't show up in the worlds list.  To import it, I simply unzipped the archive from Minefold into my ~/Library/Application Support/minecraft/saves folder.  I'm running Minecraft 1.8, and I don't have any mods installed.  There's no error; the world just doesn't show up in my list of worlds after I click [Singleplayer] (or [Multiplayer]).  Now, the big clue is that the directory structures are different between my locally-created worlds and the world created on Minefold, but I can't figure out how to adapt the files to make them match:  This is a screenshot of my saves folder. I've circled in blue "local" and "minefold", which contain data files for the local and Minefold worlds, respectively.    I bracketed similar filenames in both worlds, but I'm flummoxed in that the Minefold version has three (3) sets of data files, whereas the local world has only one (1) set of files (even though I have visited Nether in the local world).  How do I load my old Minefold world locally? Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ I would like to create a half-turn staircase without a landing. Is there an easy way to accomplish this without having to manually calculate and create each step individually?  Here’s an image similar to what I would like to achive:    I have found the Archimesh add-on including . While I can bend the created stairs, I haven’t found any way to get anywhere close to the image above. My stairs just always look out of shape and I don’t know how to get the outline of the staircase into a square.  I should mention that I only have very little experience with Blender modeling, so it’s well possible that I’m just missing some curve modification basics. Perhaps Archimesh is also the wrong approach for this in the first place? Studying for CISSP these choices seemed limited:     What is the advantage of digital signatures over message authentication codes?         Digital Signature provides integrity verification while message authentication code does not   Digital Signature provides confidentiality while message authentication code does not   Digital Signatures provide authenticity while message authentication code does not   Digital Signatures works faster than message authentication codes      Using the following chart as a reference, I'm leaning towards #2, but none of the choices really fit properly…  Cryptographic primitive | Hash |    MAC    | Digital Security Goal           |      |           | signature ------------------------+------+-----------+------------- Integrity               |  Yes |    Yes    |   Yes Authentication          |  No  |    Yes    |   Yes Non-repudiation         |  No  |    No     |   Yes ------------------------+------+-----------+------------- Kind of keys            | none | symmetric | asymmetric                         |      |    keys   |    keys   Hopefully an expert here can clarify. How many Marvel Earths (Universes) are there? Temperature and kinetic energy of molecules I was wondering if temperature is related to the average translational kinetic energy of the molecules then why does the average kinetic energy of say a moving object not affect the temperature? What is the mechanism regarding these vibrational motions that imparts temperature. Also do a gas and liquid at the same temperature possess the same kinetic energy? and a heavier molecule traveling at the same speed as a lighter molecule will have a greater temperature? We noticed that lots of bugs in our software developed in C# (or Java) cause a NullReferenceException.  Is there a reason why "null" has even been included in the language?  After all, if there were no "null", I would have no bug, right?  In other words, what feature in the language couldn't work without null?