All prime ideals are invertible $\Rightarrow$ Dedekind domain
In an integral domain every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible.
I feel stupid asking, but I can't seem to find the answer to this anywhere. I'm trying to follow  to edit my bash prompt, but there is no .bashrc in either my user directory (~/.bashrc), or in my home directory, or in the main file system directory.   There is no .bashrc in /etc/ (there is a bash.bashrc, however).   Do I create a .bashrc file in my user or home directory? Is there a better, more recent set of instructions for modifying the bash prompt? How prove $\frac{a^2}{a+2b^2}+\frac{b^2}{b+2c^2}+\frac{c^2}{c+2a^2}\ge 1$ let $a,b,c$ be postive real numbers ,and such $$ab+bc+ac=3$$   show that   $$\dfrac{a^2}{a+2b^2}+\dfrac{b^2}{b+2c^2}+\dfrac{c^2}{c+2a^2}\ge 1$$   This problem is from Secrets In Inequalities volume 1  page 30,example 1.24. the comment.the  author this case is a bit more diffcult,But this author can't post solution.Thank you How can I merge properties of two JavaScript objects dynamically? How can I install MonoDevelop v4 on 12.04? I am using Ubuntu 12.04. I can install v2.8.6.3 of Monodevelop from software center but I need at least v3 to use Monogame v3 or higher. Can anyone point me in the right direction please?   Thanks I am considering buying a Dell XPS 15 laptop. The laptop has a Nvidia 420m card which should work with Linux but I keep reading that the Nvidia Optimus technology isn't supported on Linux. I am not really interested in switching from Nvidia to Intel to save power but need to know that the Nvidia card will in fact work if I installed Ubuntu.  If anyone has experience using a Nvidia card with Optimus technology or even better the exact laptop in question (Dell XPS 15 with 1GB NVIDIA® GeForce®  GT 420M) it would be great.  A major problem holding people back from adopting Linux is this sort of hardware issue. I am a long term Ubuntu user and supporter who can't afford to make a mistake with a purchase like this. I don't want to spend £500+ to find I have no graphics acceleration because Windows 7 is not an option. How to send email from MySQL 5.1 Radius of Convergence of a given Power series Proof of this fairly obscure differentiation trick? Suppose we're tying to differentiate the function $f(x)=x^x$. Now the textbook method would be to notice that $f(x)=e^{x \log{x}}$ and use the chain rule to find $$f'(x)=(1+\log{x})\ e^{x \log{x}}=(1+\log{x})\ x^x.$$  But suppose that I didn't make this observation and instead tried to apply the following differentiation rules:  $$\frac{d}{dx}x^c=cx^{c-1} \qquad (1)\\ \frac{d}{dx}c^x = \log{c}\ \cdot c^x \quad (2)$$  which are valid for any constant $c$. Obviously neither rules are applicable to the form $x^x$ because in this case neither the base nor the exponent are constant. But if I pretend that the exponent is constant and apply rule $(1)$, I would get $f'(x)\stackrel{?}{=}x\cdot x^{x-1}=x^x.$ Likewise, if I pretend that the base is constant and apply rule $(2)$ I obtain $f'(x)\stackrel{?}{=}\log{x}\cdot x^x$.  It isn't hard to see that neither of the derivatives are correct. But here's where the magic happens: if we sum the two “derivatives” we end up with $$x^x+ \log{x}\cdot x^x=(1+\log{x})\ x^x$$ which is the correct expression for $f'(x)$.  This same trick yields correct results in other contexts as well. In fact, in some cases it turns out to be a more efficient way of taking derivatives. For example, consider $$g(x)=x^2 = \color{blue} x\cdot \color{red} x.$$ If we pretend the blue $\color{blue} x$ is a constant we would get $g'(x)\stackrel{?}{=}\color{blue}x\cdot 1=x$. Now if we pretend the red $\color{red}x$ is constant we get $g'(x)\stackrel{?}{=}1\cdot \color{red} x=x$. Summing both expressions we end up with $2x$ which is of course a correct expression for the derivative.  These observations have led me to the following conjecture:     Let $f(x,y)$ be a differentiable function mapping $\mathbb{R}^2$ to $\mathbb{R}.$ Let $f'_1 (x,y)=\frac{\partial}{\partial x} f(x,y)$ and $f'_2 (x,y)=\frac{\partial}{\partial y} f(x,y)$. Then for any $t$ we have: $$\frac{d}{dt}f(t,t)=f'_1 (t,t) + f'_2 (t,t).$$   (I apologise for the somewhat awkward notation which I could not seem to get around without causing undue ambiguity.)  This formulation also seems to lend itself to following generalisation:     Let $f:\mathbb{R}^N \to \mathbb{R}$ be a function differentiable in each of its variables $x_1,x_2,\ldots,x_N$. For $n=1,2,\ldots,N$ define $f'_n(x_1,x_2,\ldots,x_N)=\frac{\partial}{\partial x_n}f(x_1,x_2,\ldots,x_N)$. Let $t$ be any real number and define the $N$-tuple $T=(t,t,\ldots,t)$. Then one has: $$\frac{d}{dt} f(T)=\sum_n f'_n(T).$$   Thus my question is:   Is this true? How can it be proven? (Specifically in the case $N=2$ but also in the general case.) How can I define a mathematical function as a LaTeX macro? How to put small numbers outside the pie slices without making a mess I'm new to LaTeX and need some help with my figures. I would like to to put small numbers outside the pie slices without making a mess. Having found this thread , I tried it out, but didn't get all the way through with it. Right now, my pie looks like this:    I used the following code:  \usepackage{tikz,stackengine} \makeatletter  \tikzstyle{chart}= [legend label/.style={font={\scriptsize},anchor=west,align=left}, legend box/.style={rectangle, draw, minimum size=5pt}, axis/.style={black,thin,-&gt;}, axis label/.style={anchor=east,font={\tiny}}]  \tikzstyle{bar chart}=[ chart, bar width/.code={     \pgfmathparse{##1/2}     \global\let\bar@w\pgfmathresult}, bar/.style={thin, draw=black}, bar label/.style={font={\bf\small},anchor=north}, bar value/.style={font={\footnotesize}}, bar width=.75,]  \tikzstyle{pie chart}= [chart, slice/.style={line cap=round, line join=round, thin, draw=black}, pie title/.style={font={}}, slice type/.style 2 args={     ##1/.style={fill=##2},     values of ##1/.style={}}]  \pgfdeclarelayer{background} \pgfdeclarelayer{foreground} \pgfsetlayers{background,main,foreground}  \newcommand{\pye}[3][]{     \begin{scope}[#1]         \pgfmathsetmacro{\curA}{90}         \pgfmathsetmacro{\r}{1}         \def\c{(0,0)}         \node[pie title] at (270:1.3) {#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) , .5)} %%%              %\pgfmathprintnumberto{#3}{\fnum}  %\ifthenelse{\lengthtest{#3 pt &lt; 10pt}} %{\path (O) -- ++ (\midangle:\radius) %node[inner sep=0,align=left,anchor=west,rotate=\midangle](nums#3)    {~~\fnum};} %{\path (O) -- ++(\midangle:\innerpos)  %node {\scalefont{#3}\beforenumber\fnum\afternumber};}           %%%                         \begin{pgfonlayer}{foreground}                 \path \c -- node[pos=\d,pie values,values of \s]{$\v\%$} +    (\midA:\r);             \end{pgfonlayer}             \global\let\curA\nextA}     \end{scope}}  \newcommand{\legend}[2][]{     \begin{scope}[#1]         \path         \foreach \n/\s in {#2}         {++(0,-5pt) node[\s,legend box] {} +(2pt,0) node[legend label]     {\n}};     \end{scope}}      \begin{document}         \begin{center}      \begin{figure}[H]     \centering \begin{tikzpicture} [pie chart, slice type={A}{black!0}, slice type={B}{black!10}, slice type={C}{black!20}, slice type={D}{black!30}, slice type={E}{black!40}, slice type={F}{black!50}, slice type={G}{black!60}, slice type={H}{black!70}, pie values/.style={font={\small}}, scale=2]  \pye{1994}{6.1/A, 2.9/B, 0.2/C, 54.3/D, 25.2/E, 1.4/F, 6.7/G, 3.2/H} \pye[xshift=2.2cm,values of coltello/.style={pos=1.1}]% {2005}{6.8/A, 3.0/B, 0.5/C, 52.3/D, 25.6/E, 1.8/F, 7.1/G, 2.8/H} \pye[xshift=4.4cm,values of caffe/.style={pos=1.1}]% {2015}{7.2/A, 3.1/B, 0.6/C, 50.6/D, 26.5/E, 2.3/F, 7.5/G, 2.2/H}  \legend[shift={(0cm,-1.5cm)}]{{A}/A, {B}/B}  \legend[shift={(1.4cm,-1.5cm)}]{{C}/C, {D}/D}  \legend[shift={(2.8cm,-1.5cm)}]{{E}/E, {F}/F} \legend[shift={(4.2cm,-1.5cm)}]{{G}/G, {H}/H} \end{tikzpicture} \end{figure}  \end{document}   This looks quite the way I want it to, only the small numbers should be placed outside the pies.  Thanks in advance :) What is the command that lets me see what and where devices are mounted?  I'm having trouble changing songs on my old iPod, and I have a feeling it's because of the mount point. Why can't I travel onwards if my bag wasn't going to make the flight? I was flying KLM from the UK to the USA with a short layover in Amsterdam.  Because of delays in the UK, I almost missed my flight.  The flight attendant told me my bag wasn't going to make it, but if I walk fast, my gate is just two over.  When I got to the gate, they had already closed the doors, but the counter was opened - I mentioned that my bag was not going to make it, but that's okay.  She said, "hold on," got onto her walkie-talkie, and said,  "Yep, that's what I thought: you can't fly if your bag's not going."  I originally thought that it might be a security concern, but that doesn't really make any sense.  Why can't I travel if my luggage doesn't make the connection?  EDIT: I should clarify that I have no beef with KLM - they treated my overnight in Amsterdam professionally and helpfully; I'm just curious what the reason would be. Align cell height of the tabularx rows The following latex code prints rows one at a time. However all the cells of the rows don't come out aligned. How can I fix it.   \documentclass{beamer}     \usepackage{tabularx}    \begin{document}  \begin{frame}{}  \begin{tabularx}{3.50in}{lXX}           \hline           &amp; Observation &amp; Conclusion \\           \hline             \uncover&lt;2-&gt;{1.} &amp; \uncover&lt;2-&gt;{Observation 1 } &amp; \uncover&lt;2-&gt;{Conclusion 1} \\   \uncover&lt;3-&gt;{2.} &amp; \uncover&lt;3-&gt;{Observation 2} &amp; \uncover&lt;3-&gt;{Conclusion 2} \\           \end{tabularx}      \end{frame}   \end{document}   The output looks as follows :    As we can see cells in first column do not align with the rest of the rows. Will appreciate any input as to how to fix this. Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$ How to find an ellipse , given 2 passing points and the tangents at them? Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ellipse which satisfies it?  Thank you in advance.  Edit : May be , I can say the first answer (by Mr. André Nicolas) is for the general case. Is there no special case where only a finite set of ellipses satisfies the condition?  Edit : According to the answers and comments , I can compute an arbitrary chosen ellipse for my condition (using the method by Mr.Patrick Da Silva). But there are possibly many others which satisfy my conditions. Am I right? After I installed Ubuntu 13.04, I can't mount my NTFS partition.  Error mounting /dev/sda5 at /media/mrxfly/xFLyDiskData: Command-line `mount -t "ntfs" -o "uhelper=udisks2,nodev,nosuid,uid=1000,gid=1000,dmask=0077,fmask=0177" "/dev/sda5" "/media/mrxfly/xFLyDiskData"' exited with non-zero exit status 14: The disk contains an unclean file system (0, 0). Metadata kept in Windows cache, refused to mount. Failed to mount '/dev/sda5': Operation not permitted The NTFS partition is in an unsafe state. Please resume and shutdown Windows fully (no hibernation or fast restarting), or mount the volume read-only with the 'ro' mount option. Updating to latest Blender version I currently have version 2.74. How do I update to version 2.76? I do not see any links in the UI to update my version of Blender Is there any noun in English which changes the first letter in the plural? Plenty of nouns change the second letter to become plural (man->men, goose->geese) but does anything change its first letter.  I've hunted high and low over the internet, and spent ages browsing the questions at  but I can't find anything. How do I install Citrix receiver? Has anyone managed to get the Citrix receiver client (icaclient) working in Ubuntu? Terminal, how to quit --More-- list