How do I prove $2xy ≤ x^2+y^2$? I'm quite new at calculus, how could I prove:  $$2xy ≤ x^2 + y^2$$  I just can end up in $(xy)^{1/2} ≤ (x+y)/2$. And even though I know that this is true too, I don't know how to prove it either. Thank you in advance :D
How can I prove that $xy\leq x^2+y^2$? How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ?
I hear so many terms involving the word "linear". Linear function, linear equation, linear system, linear operator, linear transformation, linear mapping, linear space, linear algebra, linear electrical circuits, linear filters, linear electrical elements, linear approximation, linear optimization.  I'm getting crazy trying to understand the application of the "linear concept" to all this aspects (function, equation, mapping, system, operator, transformation, algebra, etc.) and I wish to know the one essence that is to be linear. What something has to be to be linear? If I say something is linear, what do I know for sure about that something (no matter what the something is)?       I heard a definition of linearity is by homogeinity (scaling the input results in a scaled output) and addition (summing the inputs results in summing the outputs). Can I apply this simple definition to all the branches (operator, mapping, system, transformation, algebra, ...) I mentioned ? Do they all behave like a line ?  y = ax + b, for example, is a line but doesn't behave like a line because y is not linear. Greatest common divisor of two positive integers I know that Greatest common divisor of two positive integers $a$ and $b$ is the largest positive integer that divides both $a$ and $b$, but how can I use that to prove that $\gcd(a, b) = \gcd(a, b-a)$. Why can I modify a read-only file? Short Question:  Why can we manipulate a read-only file in Vim using : + w + q + ! even without being an administrator?  Long Question:  I have a text file (myFile.txt) which is read-only for everyone:  navid@navid-ThinkPad-T530:~/ubuntuTest$ ls -l myFile.txt  -r--r--r-- 1 navid navid 26 Aug 22 21:21 myFile.txt   I can open it with Vim without having admin privileges:  navid@navid-ThinkPad-T530:~/ubuntuTest$ vi myFile.txt    I modify it and press: Esc + : + w + q + Enter and I see this error message:  E45: 'readonly' option is set (add ! to override)   So far, everything makes sense.  But when I press: Esc + : + w + q + ! + Enter, Vim saves the changes.  I'm using Ubuntu 16.04 and VIM 7.4. No processing tools in QGIS 3.2 Managing constructors with many parameters in Java In some of our projects, there's an class hierarchy that adds more parameters as it goes down the chain. At the bottom, some of the classes can have up to 30 parameters, 28 of which are just being passed into the super constructor.  I'll acknowledge that using automated DI through something like Guice would be nice, but because of some technical reasons, these specific projects are constrained to Java.   A convention of organizing the arguments alphabetically by type doesn't work because if a type is refactored (the Circle you were passing in for argument 2 is now a Shape) it can suddenly be out of order.  This question might be to specific and fraught with "If that's your problem, you're doing it wrong at a design level" criticisms, but I'm just looking for any viewpoints. Does $\sqrt{a+b} \le \sqrt a + \sqrt b$ hold for all positive real numbers a and b? I thought of this a while ago, but can't make up a proof or a counterexample. Does anyone know more about this?  $$\sqrt{a+b}  \le \sqrt a + \sqrt b , \forall a,b \in \mathbb R_+$$  Moreover, what happens with more variables? Say:  $$\sqrt{x_1+x_2+...+x_n} \le \sqrt x_1 +\sqrt x_2 + ... + \sqrt x_n $$  with $ x_i \in\mathbb R_+ \forall i \in \{1,2,...,n\} $  Or when $i = \mathbb N$  PS: I tried looking on the internet for this but I don't know how I would call this. When I launch nautilus from terminal the following error is printed and nautilus doesn't open up.  (nautilus:25179): GLib-GIO-CRITICAL **: g_dbus_interface_skeleton_unexport: assertion 'interface_-&gt;priv-&gt;connections != NULL' failed  (nautilus:25179): GLib-GIO-CRITICAL **: g_dbus_interface_skeleton_unexport: assertion 'interface_-&gt;priv-&gt;connections != NULL' failed Could not register the application: Timeout was reached  (nautilus:25179): Gtk-CRITICAL **: gtk_icon_theme_get_for_screen: assertion 'GDK_IS_SCREEN (screen)' failed  (nautilus:25179): GLib-GObject-WARNING **: invalid (NULL) pointer instance  (nautilus:25179): GLib-GObject-CRITICAL **: g_signal_connect_object: assertion 'G_TYPE_CHECK_INSTANCE (instance)' failed   What could be the problem?? Quote from :     The only portable way of using templates at the moment is to implement them in header files by using inline functions.   Why is this?  (Clarification: header files are not the only portable solution. But they are the most convenient portable solution.) Expected number of siblings, given the expectation of children in a family The number of children in a family is a random variable $X$, where    $ \mathbb E(X) = 1.8 $ and $Var(X) = 0.36 $. If we randomly choose a child, which is the expected number of its siblings?      (it is given that the answer $&gt; 0.8$)   Let the number of a child's siblings be a random variable $Y$, so we are looking for $ \mathbb E(Y) $.  If found that,  $$ \mathbb E(Y) = \sum_{k=0}^{\infty}k \; \mathbb P(Y=k) = \sum_{k=0}^{\infty}k \; \mathbb P(X=k+1) = $$ $$ = \sum_{k=0}^{\infty}k \; \mathbb P(X=k)-\sum_{k=0}^{\infty}\mathbb P(X=k) +\mathbb P(X=0)= \mathbb E(X)-1+\mathbb P(X=0) $$ which is indeed greater than $0.8$, if  $\; \mathbb P(X=0)&gt;0$.  But, I don't know how to make use of the variance to find the final answer.  Thank you in advance Is there a way to transfer more than one Pokemon at a time? Is there a way to transfer more than one Pokemon at a time to the professor?  Or possibly an addon that gives me this ability?  Vendoring all the trash Pokemon for candies individually is really annoying and time consuming. I'm a happy Minecraft player. I place my blocks, mine my coal, and generally nobody gets hurt. Lately though, those mean creepers have been sneaking up on me from behind, and killing me. Why? Because my framerate is so darn low.  Adjusting the graphics option (Fancy to Fast) and the render distance help, but they are not enough. I have seen mention of outside optimizations and programs that people have made/used.  What hacks, tricks, and mean mods can I use to improve my Minecraft FPS? I've seen references to curried functions in several articles and blogs but I can't find a good explanation (or at least one that makes sense!) How has the age of the Universe been derived from the observations made by the Planck mission? Diagonalizable matrix with parameters $$     \begin{bmatrix}     -2 &amp; -2 &amp; 0 &amp; 0 \\     -b &amp; -b &amp; 0 &amp; 0 \\      0 &amp; 0 &amp; -4 &amp; 4 \\      0 &amp; 0 &amp; b &amp; -b \\     \end{bmatrix} $$  I have this matrix and I need to find the values for $b$ that this matrix won't  be diagonalizable.  For what I got this matrix is diagonalizable for every $b$ value.  Please write me also the final answer, thank you very much. I am studying different point estimate methods and read that when using MAP vs ML estimates, when we use a "uniform prior", the estimates are identical. Can somebody explain what a "uniform" prior is and give some (simple) examples of when the MAP and ML estimators would be the same? I have just asked a question on   I read from this meta answer  that suggested     From a desktop support perspective there are certainly cases where a   large(r) number of machines will be affected by something, in which   case it's appropriate for SF. There are then others that are isolated   to a single computer, these tend to be the ones I vote to move to SU.   Since my question is clearly not about a single computer and is external to a single computer, I turned to ServerFault. I thought the question is about how to replicate a network admin strategy on my own (home) network. My question is, however, marked as off-topic, and I did see that homenetwork questions are off-topic. But this is not a single computer issue either.  My question is:  Where should such home/small network questions be asked?  What's the real difference between "professional" networks and the smaller networks that I build for my home or lab with a handful computers?  -- UPDATE --  The question asked here is different from the said duplicate question  as that question is about the difference among computer science and programming SE sites. It has nothing to do with network administration, whether it's large or small networks. None of the two SE sites in question (Server Fault and Super User) is even mentioned in the question or in the 2 cited answers.  -- UPDATE2 --  This question is not a duplicate of , which was cited in the OP and was the cause of this question. That question does not cover the overlapping area of small networks, which could both be for work and for home network. In the beginning of the series, it's mentioned that Muggle-born wizards/witches are identified due to inadvertent spell-casting. However, it seems like later in the book, disarmed wizards/witches are completely helpless.   What is the explanation for why some spells can be cast without a wand and is there an in-universe explanation for why this isn't done? Also, are there spells that can be cast non-verbally (with/without a wand)? Convergence of $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$ - central limit theorem Suppose that $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$,  where $X_1,X_2,\ldots$ and $Y_i,Y_2, \ldots$ are i.i.d. sequences of random variables with mean $\mu$ and variance $\sigma^2$ and the sequences $\{X_i\}$ and $\{Y_i\}$ are independent.  Then, for all real numbers $x$, show that, as $n\to \infty$,  $$\mathbb{P}(U_n\leq x) \to  \int_{-\infty}^{x} \frac{1}{(\sqrt(2\pi))} \:e^{-\frac 12{u^2}}\mathrm du $$ Approach  I tried to carry out the integration on the right, but this does not integrate nicely. It ends up involving an error function, so I figured this was the wrong was to go.  Should I be looking at convergence in probability to answer this question? Is Chebyshev's inequality relevant here? If so, how and how does that relate to the integral? I'm getting this error:  ! Undefined control sequence. l.64 ...p(- \frac{E_j - E_i}{k_B T}) &amp; \quad \text                                                   {:  E_j &gt; E_i}\\ ?  ! Undefined control sequence. l.65     1 &amp; \quad \text                         {:  E_j&lt;E_i}\\ ?    which is from this:  \begin{equation}\label{eq:hop} w_{i,j}=w_0 exp(-2 \gamma R_{i,j})\left\{    \begin{array}{l l}     exp(- \frac{E_j - E_i}{k_B T}) &amp; \quad \text{:  E_j &gt; E_i}\\     1 &amp; \quad \text{:  E_j&lt;E_i}\\   \end{array} \right. \end{equation}   But I don't understand why I get the error. This is my header:  \documentclass[11pt]{iopart} %\newcommand{\gguide}{{\it Preparing graphics for IOP journals}} %Uncomment next line if AMS fonts required \usepackage{iopams} \usepackage{graphicx, wrapfig} \usepackage{fixltx2e} \graphicspath{{images/}} \DeclareGraphicsExtensions{.pdf,.png,.jpg,.gif} \makeatletter \g@addto@macro\@floatboxreset\centering %\def\@cite#1{\textsuperscript{[#1]}} \makeatother   And my class file is the standard IoP one from   Can anyone help me get read of this error? Should I remove Adult Dependent from my 2019 tax return so they will receive a COVID Stimulus Check? On my 2019 Tax Return I claimed my 28 year old girlfriend as a dependent. This makes her ineligible to receive a COVID Stimulus Check. My return was already accepted and I have already received my refund. Should I amend my return, remove her as a dependent? Would she then be eligible to receive the stimulus check?