Does there exist a GCD domain $R$ in which the relationship $aR \cap b R \subseteq \mathrm{lcm}(a,b) R$ is untrue?
Showing that $(m)\cap (n)=(\operatorname{lcm}(m,n))$ and $(m)+(n)=(\gcd(m,n))$ for any $m,n\in\mathbb{Z}$
Why I cannot use universal instantiation rule here? I have the following premise:      $$\forall x (Fx \vee \neg Fx) \rightarrow Ga$$   Why I cannot use UI and get: $(Fa \vee \neg Fa) \rightarrow Ga$ And what is the difference between the first premise and the below premise:     $$\forall x ((Fx \vee \neg Fx) \rightarrow Ga)$$   Thanks in advance. Energy conservation in General Relativity Booting Ubuntu Failure : error: attempt to read or write outside of disk 'hd0' I have installed Ubuntu 12.10 in a Western Digital external hard drive (320GB). This is a complete installation, not a live USB.  When I plug it in my HP desktop I go to the BIOS settings and boot off the hard drive, everything work perfectly as it should. Now this works on every single computer and laptop in my house (all HP), except for ONE. My HP ProBook 4530s.  When I select to boot of the USB I get the message:  error: attempt to read or write outside of disk 'hd0'   Now, I have removed the hard drive from my laptop and the external drive is the ONLY drive plugged in. Below is a screenshot of the message on the screen. After the message I navigate to ls / (as shown below):    After here I try to acces other folders under ls /, for example, I try to go to ls /boot to get to the grub folder. Then I get the same message as before: as shown below:    grub rescue&gt; ls /boot  error: attempt to read or write outside of disk 'hd0' grub rescue&gt; _   The only folders I can access without getting the message again are /home, /run and /usr.   So how do I:   Boot Ubuntu from GRUB2 (this screen) manually Set to automatically boot Ubuntu If possible an explanation for this problem   Thanks! Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$ Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$;  I was stuck with this question for a while... Help me please!!! Thanks!!! Google street view URL Hi I have the address of a property and my application can launch a browser to go to . If a good match is found it will take it directly there. Is there any thing I can append to the url to make it switch to streetview without having the exact coordinates? I dont't want to code any javascript or flash. ref to subsection number only is there a way to reference the subsection part only? Example:  \documentclass{article} \begin{document}  \section{This is my section} For further information see subsection \ref{subsecthree} within this section. \subsection{One} \subsection{Two} \subsection{Three}\label{subsecthree}  \end{document}   Usually gives (without the stars):  1 This is my section For further information see subsection **1.3** within this section.   But I want:  1 This is my section For further information see subsection **3** within this section.   Is that possible?  [Edit] To make it more clear: I am looking for a command that works like \ref, but prints (the arabic number of) the value of the subsection counter at the position of the \label.  ======================= == Solution in LaTeX ==  There is another solution below. This is a different solution I came up with  (I cannot post an answer to my own question, so I put it here):  \documentclass{article}  \newcommand{\labelsubseccounter}[1]{     \renewcommand\thesubsection{\arabic{subsection}}     \addtocounter{subsection}{-1}     \refstepcounter{subsection}     \label{#1}     \renewcommand\thesubsection{\thesection.\arabic{subsection}} }  \begin{document}  \section{This is my section} For further information see subsection \ref{subsecthree} within this section,  which is called \ref{onepointthree}, not \ref{onepointfour}. \subsection{One} \subsection{Two} \subsection{Three}\label{onepointthree}\labelsubseccounter{subsecthree} \subsection{Four}\label{onepointfour}  \end{document}   (I would like to insert the compiled code, but don't know how, sorry) Is it possible now to use microtype with opentype fonts? Add a linebreak in an HTML text area How can i add a line break to the text area in a html page? i use VB.net for server side coding. So I was thinking about something for the past while  Consider a large spherical foam-ball with homogeneous density. Where a foam ball is defined as an object that can absorb matter with 0 friction (example: a gravitational well without an object inside). This is a purely theoretical construct.  If the foam-ball has a radius R and a charge Q. What charge must the foam-ball have, such that there is well defined sphere or horizon such that any object with a negative charge (even if it is the slightest bit) must travel faster than C in order to escape the field of the ball.  Aka what charge would turn this into a black-hole for all objects with opposite sign on their charge?  Once again I assume the foam-ball is a single particle, that doesn't repel itself... its just a very large homogenous 'thing' with charge. I have a query going that gets data for an ID for the last 3 months.  I need to tweak it so I get the highest value for each of the three months.  I've tried a couple of things with the aggregate function MAX, but I'm not getting anywhere.  I'm trying to get the max value for each of the past months ....  Here's the data from the query, currently sorted by date (asc):   ID      Date               Value 12410   01/03/2017 12:17    0.000178 12410   01/10/2017 11:36    0.000186 12410   01/17/2017 11:27    0.000189 12410   01/24/2017 13:09    0.000182 12410   01/31/2017 10:37    0.000169 12410   02/07/2017 11:03    0.000214 12410   02/14/2017 11:52    0.000176 12410   02/21/2017 10:51    0.000200 12410   02/28/2017 12:29    0.000194 12410   03/07/2017 08:39    0.000206   Here's the query:   select AnalysisID as "ID" , AnalysisDateTime as "Date", AnalysisValue as "Value" from AnalysisValueTbl where AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -3, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-3, getdate())) or AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -2, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-2, getdate())) or AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -1, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-1, getdate())) order by AnalysisValue desc What's the difference between JavaScript and Java? What's the difference between JavaScript and Java? I have an HTML string representing an element: '&lt;li&gt;text&lt;/li&gt;'. I'd like to append it to an element in the DOM (a ul in my case). How can I do this with Prototype or with DOM methods?  (I know i could do this easily in jQuery, but unfortunately we're not using jQuery.) Why is Dictionary preferred over Hashtable in C#? In most programming languages, dictionaries are preferred over hashtables. What are the reasons behind that? Mathematica envelope for the bottom of a plot, a generic function Matrix Multiplication in context of row and column vectors $\int_0^\infty \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx$ Trying to compute   I was facing: \begin{align}J=\int_0^\infty \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx\end{align}  I want to prove that $\displaystyle J=0$, or equivalently, that,  \begin{align}\int_0^1 \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx=-\dfrac{1}{2}\text{G}\end{align}  $\text{G}$ being the Catalan constant.  Read carefully please.   I know, using so-called Feynman's trick, how to prove this. I would like to obtain a proof, using only integration by parts and change of variable in simple integrals (that is, no multiple integrals) I don't know, if, under these restrictions, such computation is possible.  NB: You're probably wondering what is the link between : $\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx$ and $J$.  The link is, for $x\in\left[0;\frac{\pi}{2}\right]$, $(\sin{x}+\cos{x}+\sqrt{\sin{2x}})(\sin{x}+\cos{x}-\sqrt{\sin{2x}})=1$ and $\sin(2x)=2\sin x\cos x$ Can you wait to add optional damage dice that apply on a hit (such as Divine Smite) until after seeing if the roll is a critical? I want Ctrl+Alt+B bring up a browser.  Ctrl+Alt+Delete is handy.  Can I set up Ctrl+Alt+B to bring up a browser (either default or specific)? How can I prevent 'grep' from showing up in ps results? When I  search for some process that doesn't exist, e.g.  $ ps aux | grep fnord                           wayne    15745  0.0  0.0  13580   928 pts/6    S+   03:58   0:00 grep fnord   Obviously I don't care about grep - that makes as much sense as searching for the ps process!  How can I prevent grep from showing up in the results? Where do I find the current C or C++ standard documents? For many questions the answer seems to be found in "the standard". However, where do we find that? Preferably online.  Googling can sometimes feel futile, again especially for the C standards, since they are drowned in the flood of discussions on programming forums.  To get this started, since these are the ones I am searching for right now, where are there good online resources for:   C89 C99 C11 C++98 C++03 C++11 C++14 C++17