Proving inequalities hold for all values
Proving $\frac{x}{y} +\frac{y}{z} + \frac{z}{x} \ge 3$ for positive $x,y,z$
Is there a streaming API for JSON? How to print object content in correct way? Error notification on plone 4 Why am I asked for a credit card number to download a free app? I  want to download free app in my iPod touch device and i have the following doubts.  Kindly resolve me doubts.   when i try to download it asks my credit card security number. since i am trying to download free app , why do i need to provide my credit card security code? Will i be debited for downloading the free app from app store?   I cant able to find any information regarding this on apple.  Please provide me some references about this from apple and please resolve my doubts.  Thanks a lot. Pie chart labels I want to create a pie chart with labels (one blank an one with percentage for the teacher, the mwe is the teachers version) so that my students have to fill the percentage in by themselves.  So far I created a pie chart using the code given in the answer here    and then I used labels from the code given in the answer here    Now I have this    The node in the violett part is the one I created manually and the other ones are implemented in the code, but I don't know what to change so that the automatic generated ones look like the one I created manually  My MWE      \documentclass[11pt]{article}   \usepackage{tikz} \usetikzlibrary{arrows}  \usepackage{xcolor}  \usepackage[utf8]{inputenc}  \definecolor{rosso}{RGB}{220,57,18} \definecolor{giallo}{RGB}{255,153,0} \definecolor{blu}{RGB}{102,140,217} \definecolor{verde}{RGB}{16,150,24} \definecolor{viola}{RGB}{153,0,153}  \makeatletter  \tikzstyle{chart}=[     legend label/.style={font={\scriptsize},anchor=west,align=left},     legend box/.style={rectangle, draw, minimum size=5pt},     axis/.style={black,semithick,-&gt;},     axis label/.style={anchor=east,font={\tiny}}, ]  \tikzstyle{pie chart}=[     chart,     slice/.style={line cap=round, line join=round, very thick,draw=white},     pie title/.style={font={\bf}},     slice type/.style 2 args={         ##1/.style={fill=##2},         values of ##1/.style={}     } ]  \pgfdeclarelayer{background} \pgfdeclarelayer{foreground} \pgfsetlayers{background,main,foreground}   \newcommand{\pie}[3][]{     \begin{scope}[#1]     \pgfmathsetmacro{\curA}{90}     \pgfmathsetmacro{\r}{1}     \def\c{(0,0)}     \node[pie title] at (90:1.3) {#2};     \foreach \v/\s in{#3}{         \pgfmathsetmacro{\deltaA}{\v/100*360}         \pgfmathsetmacro{\nextA}{\curA + \deltaA}         \pgfmathsetmacro{\midA}{(\curA+\nextA)/2}          \path[slice,\s] \c             -- +(\curA:\r)             arc (\curA:\nextA:\r)             -- cycle;         \pgfmathsetmacro{\d}{max((\deltaA * -(.5/50) + 1) ,1)}           \begin{pgfonlayer}{foreground}         \draw[*-,thin] \c -- (\midA:\r) node[above,near end, pos=\d,pie values,values of \s]{$\v\%$};         \end{pgfonlayer}               \global\let\curA\nextA     }     \end{scope} }  \newcommand{\legend}[2][]{     \begin{scope}[#1]     \path         \foreach \n/\s in {#2}             {                   ++(0,-10pt) node[\s,legend box] {} +(5pt,0) node[legend label] {\n}             }     ;     \end{scope} }   \begin{document}   \begin{tikzpicture} [     pie chart,     slice type={comet}{blu},     slice type={legno}{rosso},     slice type={coltello}{giallo},     slice type={sedia}{viola},     slice type={caffe}{verde},     pie values/.style={font={\small}},     scale=2 ]  \pie{Sommerjobs in Österreich}{52/comet,12/legno,20/sedia,16/coltello}   \legend[shift={(-1cm,-1cm)}]{{Eisverkäufer}/comet, {Schwimmbad}/legno, {Zettelverteiler}/coltello} \legend[shift={(1cm,-1cm)}]{{Kellner}/sedia}     \draw [*-,thin] (0.6,-0.2)  -- (0.7,-0.2 + 0.5ex) -- (2,0.5ex) node[above, near end]{20\%};    \end{tikzpicture}   \end{document} I am seeking advice on which memory to use with my motherboard and CPU. Below are my components and some relevant information extracted from their official websites.   NOTE: Although this is a very specific case, a general answer is equally good for me — and also alot more useful for everyone else!  CPU: AMD Phenom II X4 810   ASUS M4A785TD-V EVO listed as one of many  motherboards.   Motherboard: ASUS M4A785TD-V EVO   4 x DIMM, Max. 16 GB, DDR3 1800(O.C.)*/1600(O.C.)/1333/1066/800 ECC,Non-ECC,Un-buffered Memory Dual Channel memory architecture (Complete list of specs: ).   NOTE II: I wouldn't want to do any overclocking as I have no experience with that. So I think the 1333 MHz is the most appropriate. Since the higher frequencies are marked with "O.C."  Questions:   How can I be certain that a specific RAM will work well (or not work) together with my CPU and motherboard? (The Qualified Vendor List is a good starting point, but it's a bit limited and sometimes a bit outdated). Is it true that choice of memory is not independent of the processor? How?    I'm worried I'll be missing something essential when buying my next RAM chips :) How can you make an armor that has infinite protection armor? When I was using the new attribute command, I saw an attribute called &quot;armor.&quot; I don't know what it actually does, but I think it has to do with protection. I was wondering if I could make an armor that has infinite protection so I tried to make a command: give @s netherite_helmet{AttributeModifiers:[{AttributeName:&quot;generic.armor&quot;,Name:&quot;generic.armor&quot;,Amount:1.79769313486232E+308d,Operation:0,UUID:[I;439,684,724,99]},Slot:&quot;head&quot;}]}  I thought it would give me infinite protection, but it only gives me a helmet with the empty &quot;protection&quot; attribute. It works fine on older versions, but not on the 1.17 snapshots. These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1:  Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=15, \ LCM(x,y)=150$ $b) \ GCD(x,y)=120 \ LCM(x,y)=1320$ $c) \ GCD(x,y)=100 \ LCM(x,y)=990$  Exercise 2:   Find all the numbers $m,n$ such that $GCD(m,n)=pq , \ LCM(m,n)=p^2qs$  where $p,q,s$ are prime  The first thing that is known to me is that $GCD(x,y) \cdot LCM(x,y)= x \cdot y$ Also $LCM(x,y)$ is at most $x \cdot y$ while $GCD(x,y)$ is at most $\max \{x,y\}$. Last thing is that $GCD(x,y)|LCM(x,y)$. Using all this I tried to solve the first exercise: $a)$ First two obvious pairs are $x=15, y=150$ and $y=15, x=150$. Now neither of the numbers can be bigger than $150$ or smaller than $15$. So we are looking for numbers in the range $15-150$ that satisfy $x \cdot y = 15 \cdot 150$ Another such pair is $(x,y)=(30,75), \ (x,y)=(75,30)$. Similarly for $b)$ we find that the only possible values are permutations of the set {$120,1320$} and in $c)$ since $100$ does not divide $990$ no such numbers exist. Now exercise 2 is what made me think there is actually another connection I'm not quite aware of since now it's about arbitrary prime numbers and the previous method doesn't work anymore. My intuition is that it has something to do with $GCD$ or $LCM$ of the $GCD(x,y), \ LCM(x,y)$ Why was the MCU Super Soldier program considered a failure? In Marvel Cinematic Universe, military tried to create super soldiers like Hulk, but they failed. So, the program was shut down.  Where did they fail? Didn't Blonsky get Wolverine-type super healing capability? Why couldn't it make the military stronger? Other than this, wasn't Abomination a success (despite the fact that it wasn't an original work of the military)? He was defeated by Hulk, but enemy forces don't have a Hulk. When it comes to anger, who cares as long as you are focused on the target? Abomination was indeed in full control and he was always after Hulk. Intuitive explanation of unit root How would you explain intuitively what is a unit root, in the context of the unit root test?   I'm thinking in ways of explaining much like I've founded in .  The case with unit root is that I know (little, by the way) that the unit root test is used to test for stationarity in a time series, but it's just it.   How would you go to explain it to the layperson, or to a person who has studied a very basic probability and statistics course?  UPDATE  I accepted whuber's answer as it is what most reflect what I asked here. But I urge everybody that came here to read Patrick's and Michael's answers also, as they are the natural "next step" in understanding the Unit Root. They use mathematics, but in a very intuitive way. Prove the map sending a matrix to it's inverse is continuous. Should I use "who" or "that" when referring to an organisation? I generally use the rule of thumb of using who when referring to a person and that when referring to an object. Example:  Jim is someone who makes me smile. I was nearly knocked out by the ball that hit me.  But what about when we're talking about an organisation?  I am forever indebted to the charities who helped me. I am forever indebted to the charities that helped me.  The former seems somewhat warmer, if that makes sense, although the latter feels more correct. What should I use? And how about this:  I'm really happy with the insurance company who helped me with my claim. I can't stand them, they're the insurance company that ripped me off.  To me the former evokes a team of helpful people while the latter evokes a cold heartless corporate entity. Is this just a matter of style and so should go with my instincts, or should I aim for what is 'correct?' I came to know about , it's a website that allow search for flights by country. I mean you just enter from like "Germany" and to like "Belgium" and you get best offers from all airports between the two countries. That is great. And I was able to arrange a good-price trip to Brussels.   Now I want to arrange another trip to Turkey, but Whichbudget is not helping much this time. And I can't find any other websites that allow "search by country".   Update: Not duplicate to  as his question is broad to the extent of a state. But my question is broad to the extent of a whole country + less price + "Any Time". If anyone checks , he'll understand what I'm saying. Especially that whichbudget allows "Any Time" option, not just +/-3 days.  Does anyone know more similar websites? Are we close to a pure DC (or hybrid) household? This is a philosophical question. Over the last few years, many of the household devices, particularly those that are more mobile and etc are DC devices. Are we at some point where a DC household may become a possibility and Edisons dream might come true??  The array of DC devices    Internet stuff - routers, modems Most complex computer electronics - Desktops, laptops, NAS, phones, tablets, gaming consoles, etc Lighting - LED More that I cant think of or perhaps know which also convert AC -> DC   Problems with AC     Each device needs its own power supply ( AC-> DC power brick). This is a) Unpleasant to look at and deal with, and b) a costly point of failure.  Each power brick may or may not be optimized, as far as power conversion goes.   Alternative - A central (or a few ) high capacity power adapter which converts the AC into a DC power grid for the house which can then be tapped directly by the various devices. This eliminates the need for the power brick, and hopefully the central power adapter is of state of the art, and can be upgraded seperately from the devices, as needed.  Need standardization of:   Device voltage. Currently devices use 5/ 9 / 12 or others I may not be aware of, for this idea to work they need to work with one particular one, unless switching between voltages is cheap enough, or multiple DC lines are provided. Power socket input. Each device appears to have its own input socket type, this should standardize if the power supply cable should be comoditized, perhaps USB - C will play a part here. Financial proof to get a Schengen visa to France I am planning to apply for a Schengen visa to visit France. I have an attestation d'acceuil, but the visa requires I submit bank statement for the past 3 months. I can get this statement, but the problem is that my company has been paying my salary in cash for the past months, and I place the money into the account by myself.  Can I submit the following?   my accounting book (current account)  letter from the bank stating that I have money in it payslips for the past 3 months   Will that satisfy the financial requirements for the visa? Hello,  I'm getting this error every time when I boot my computer. My Ubuntu installation is fresh and I was got a few error on last step while install Ubuntu (because i have problem with my CDROM driver). Anyway, installation is continued .   For example when i tried to open ADDITIONAL DRIVER and LANGUAGE SUPPORT (on System Settings) nothing happened, it's not opening.  When i try this in terminal gnome-language-selector i'm getting this error  Traceback (most recent call last):   File "/usr/bin/gnome-language-selector", line 8, in &lt;module&gt;     from LanguageSelector.gtk.GtkLanguageSelector import GtkLanguageSelector   File "/usr/lib/python2.7/dist-packages/LanguageSelector/gtk/GtkLanguageSelector.py", line 34, in &lt;module&gt;     from aptdaemon.gtk3widgets import AptProgressDialog   File "/usr/lib/python2.7/dist-packages/aptdaemon/gtk3widgets.py", line 61, in &lt;module&gt;     class AptStatusIcon(Gtk.Image):   File "/usr/lib/python2.7/dist-packages/aptdaemon/gtk3widgets.py", line 66, in AptStatusIcon     def __init__(self, transaction=None, size=Gtk.IconSize.DIALOG): AttributeError: type object 'GtkIconSize' has no attribute 'DIALOG'   Also getting this error for jockey-gtk  Traceback (most recent call last):   File "/usr/bin/jockey-gtk", line 418, in &lt;module&gt;     sys.exit(u.run())   File "/usr/lib/python2.7/dist-packages/jockey/ui.py", line 461, in run     self.ui_show_main()   File "/usr/bin/jockey-gtk", line 81, in ui_show_main     col_icon.set_sizing(Gtk.TreeViewColumnSizing.AUTOSIZE) AttributeError: type object 'GtkTreeViewColumnSizing' has no attribute 'AUTOSIZE'   I'm worrying about are there any lack . I don't know what's wrong with my installation.  Are there any way, check my installation (packages, files etc..) . The square of an integer is congruent to 0 or 1 mod 4 Why does Windows only show about 3.5 GB of my 4 GB of RAM? I recently upgraded my computer's RAM to 4&nbsp;GB. My 32-bit Windows installation shows only 3574&nbsp;MB of the memory. How can I make Windows use the full amount of RAM? Show that an algebraically closed field must be infinite.    Answer  If F is a finite field with elements $a_1, ... , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X - a_i)$ has no root in F, so F cannot be algebraically closed.  My Question  Could we not use the same argument if F was countably infinite? Couldn't we say that if F was a field with elements $a_1, a_2, ... $ then the polynomial $f(X) = 1 + \prod_{i=1}^{\infty} (X - a_i)$ does not split over F?  Thank you in advance Can someone please explain the difference between the protected and protected internal modifiers in C#? It looks like their behavior is identical.