Is there a way of recording keystrokes while editing?
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proving logical equivalence $(P \leftrightarrow Q) \equiv (P \wedge Q) \vee (\neg P \wedge \neg Q)$ I am currently working through Velleman's book How To Prove It and was asked to prove the following  $(P \leftrightarrow Q) \equiv (P \wedge Q) \vee (\neg P \wedge \neg Q)$  This is my work thus far  $(P \to Q) \wedge (Q \to P)$  $(\neg P \vee Q) \wedge (\neg Q \vee P)$ (since $(P \to Q) \equiv (\neg P \vee Q)$)  $\neg[\neg(\neg P \vee Q) \vee \neg (\neg Q \vee P)]$ (Demorgan's Law)  $\neg [(P \wedge \neg Q) \vee (Q \wedge \neg P)]$ (Demorgan's Law)  At this point I am little unsure how to proceed.   Here are a few things I've tried or considered thus far:  I thought that I could perhaps switch some of the terms in step 3 using the law of associativity however the $\neg$ on the outside of the two terms prevents me from doing so (constructing a truth table for $\neg (\neg P \vee Q) \vee (\neg Q \vee P)$ and $\neg (\neg P \vee \neg Q) \vee \neg (P \vee Q)$ for sanity purposes)  I can't seem to apply the law of distribution since the corresponding terms are negated.  Applying demorgans law to one of the terms individually on step 2 or 3 doesnt seem to get me very far either.  Did I perhaps skip something? Am I even on the right track? Any help is appreciated In English, words with a 'g' followed by a front vowel (e, i, y) can be pronounced with a soft g or a hard g:   Words with Germanic roots are usually pronounced with a hard g: gear, get, gift, give Words with Latin and Greek roots are usually pronounced with a soft g: gem, general, giraffe, giant   But how should a purely English word (if such thing even exists) with a 'g' followed by a front vowel be pronounced? In other words, if an English speaker saw a new word of unknown origin (eg: a neologism) that starts with gi- or ge-, how would they pronounce it? Calculate battery life How can I calculate what the effect of an extension tube will be? Why is there a white space on the left side?  \documentclass{standalone} \usepackage{asymptote} \begin{document} \noindent \begin{asy} size(10cm,10cm); draw((0,0)--(100,100)); draw((0,100)--(100,0)); dot((50,50)); \end{asy} \end{document} Best way to add KeyEvents and faster Quit How can I create a Windows bootable USB stick using Ubuntu? Cardinality of a totally ordered union Let $(X_{\alpha})_{\alpha \in A}, (Y_{\alpha})_{\alpha \in A}$ be families of sets such that for $\{X_{\alpha} \mid \alpha \in A \}$ and $\{Y_{\alpha} \mid \alpha \in A \}$ are totally ordered by inclusion. We have $(X_{\alpha},Y_{\alpha}) \neq (X_{\beta},Y_{\beta})$ for $\alpha \neq \beta$. Suppose that $|X_{\alpha}| \leq |Y_{\alpha}|$ for any $\alpha \in A$. It is true that $|\bigcup_{\alpha} X_{\alpha}| \leq |\bigcup_{\alpha} Y_{\alpha}|$? In the double-slit experiment of electrons (observed by photons), is it correct to say the collapse is caused by the momentum of the photons? I'm working off the article, . I think it's well-written, but I'm not convinced about this part:     So what is causing the change in the particle’s behaviour?      In the example given above, when electrons are fired at a fluorescent screen, when we bombard the electrons with photons the interaction changes the state of the system catastrophically as, despite lacking mass, photons do carry momentum.   Here the author strongly implicates the momentum of photons as being responsible for the collapse. But is that correct to say?  I have read that the  verified that obscuring which-way information restores wave-like behavior. So it feels to me it's more about some "outer" state (perhaps well-characterized as "information") that we don't yet understand that collapses the system, not per se the photon-electron interaction.  Or am I wrong, and in fact for example, the process of obscuring in the quantum eraser experiments somehow "undoes" the photon-electron interaction or restores the quantum state through another interaction? zero matrix to the power of 0 Why $0^0=I$?   I'd tried prove that considering $N^0$ where N is a Nilpotent matrix and then using the Cayley -Hamilton theorem  Thanks in advance. EDIT:  It seems that I've also had edits that stacked well, so I'm not sure if I had a timing problem, or was just delusional.  The one thing that can be fixed, that I'll turn this question into, is a feature request so that when I go to re-edit (either within the 5-minute grace period, or before a suggested edit is approved), it brings up the old edit summary text.  Right now, it brings up a blank edit summary, and this replaced the first one I did.  This isn't really intuitive, and even knowing that's what it's doing, I still forget to put everything in.  Original Question:  It has happened a few times now, when on a site where I don't have edit privileges, that I make an edit.  I then see that the edit didn't work quite right, so I go back in to re-edit it.  This creates a brand new edit review, with an additional useless summary. How about we have the same grace period to re-edit your up-for-review edit, as there is for a normal edit? Of course, it wouldn't be possible once it's been approved. System.currentTimeMillis() vs. new Date() vs. Calendar.getInstance().getTime() In Java, what are the performance and resource implications of using  System.currentTimeMillis()    vs.   new Date()    vs.   Calendar.getInstance().getTime()   As I understand it, System.currentTimeMillis() is the most efficient. However, in most applications, that long value would need to be converted to a Date or some similar object to do anything meaningful to humans. Possible Duplicate:           What book would you recommend for an undergraduate wanting to learn the basics of topology? I've come across several books that claim to give an introduction to the subject, but I wasn't sure which one(s) to pick.  Here are some of the books I came across:   Introduction to Topology - Bert Mendelson Basic Topology - M.A. Armstrong First Concepts of Topology - William G. Chinn Topology Without Tears - Sidney A. Morris Topology - James Munkres   They don't go over the exact same topics, some are more in-depth than others, some seem to target slightly different audiences etc. What would be right for me (not necessarily from that list)? To give some more background, I want to learn topology because I feel it will be somewhat helpful in the field of Computer Vision. c++ get registry key returns only one char I have been attending a number of academic / professional conferences recently and one thing that really awed me was the flawless and natural way in which the speakers presented on their topics.  Sure, they had the occasional mispronunciations or awkward pauses but they spoke with authority and confidence.   Some of them used powerpoint slides but were not reading from the slides; they were just talking on the points and they had so much to say on each point that I felt they could not do it without notes, but they were not even looking at their notes most of the time.  (The advantages of not looking at their notes too much was that they could maintain eye contact and could use very interesting slides that were not crammed with the dot points of their speeches.)  I hate to think that they had memorised their speech notes but some of the presentations went for more than an hour.  I know rehearsal is important but I wonder how could anyone remember so much in a nervous situation. (Its one thing to know everything on your topic; its another thing to present that 'everything'!)  Question:  What background / foreground things do presenters do that make their presentations flawless and natural? Why is there a controversy on whether mass increases with speed? Some people say that mass increases with speed, some people say that the mass of an object is independent of its speed.  I understand how some (though not many) things in physics are a matter of interpretation based on one's definitions. But I can't get my head around how both can be 'true' is any sense of the word.  Either mass increases or it doesn't, right?  Can't we just measure it, and find out which 'interpretation' is right? e.g.: (in some sophisticated way) heat up some particles in a box and measure their weight?  ------------------------------  UPDATE  Right, so I've got two identical containers, each with identical amounts of water, each on identical weighing scales, and each in the same g field. If one container has hotter water, will the reading on its scale be larger than the other? If the answer is yes, and g is constant, does this mean that the m in w=mg has got bigger? Showing that two topologies give rise to the same topology How to recursively download a folder via FTP on Linux I'm trying to ftp a folder using the command line ftp client, but so far I've only been able to use 'get' to get individual files. Enqueuing Google Web Fonts the usual way, i.e., using the wp_enqueue_style function like so...  function wpse_google_webfonts() {     wp_enqueue_style( 'google-webfonts', 'http://fonts.googleapis.com/css?family=Ubuntu+Condensed|Open+Sans:400italic,700italic,400,700' ); } add_action( 'wp_enqueue_scripts', 'wpse_google_webfonts' );   ...results in a link tag placed in the header like so:  &lt;link rel='stylesheet' id='google-webfonts-css'  href='http://fonts.googleapis.com/css?family=Ubuntu+Condensed%7COpen+Sans%3A400italic%2C700italic%2C400%2C700&amp;#038;ver=3.5' type='text/css' media='all' /&gt;   As you can see, the resultant URL is encoded.  I am pretty sure it poses no problems, but to keep things clean and clear, I would like to go ahead and ask -- Is there a way to enqueue Google Web Fonts (via functions.php and not a plugin) in a way that the URL output is not encoded?  That's like so:  &lt;link rel='stylesheet' id='google-webfonts-css'  href='http://fonts.googleapis.com/css?family=Ubuntu+Condensed|Open+Sans:400italic,700italic,400,700?ver=3.5' type='text/css' media='all' /&gt;   Reason For Bounty  @webaware's answer is near perfect, especially because it's similar to the method employed in  theme.  The only problem in the output is that it's like this:  &lt;link rel='stylesheet' id='twentytwelve-fonts-css'  href='http://fonts.googleapis.com/css?family=Ubuntu+Condensed|Open+Sans:400italic,700italic,400,700&amp;#038;subset=latin,latin-ext' type='text/css' media='all' /&gt;   Notice the &amp;#038;? It should be &amp;, otherwise the font-files served only have the latin glyphs (i.e. the subset parameter in the URL is neglected unless you use &amp; and NOT its HTML entity).  Anyone who can help modify @webaware's answer so that the output's looks like this...  &lt;link rel='stylesheet' id='twentytwelve-fonts-css'  href='http://fonts.googleapis.com/css?family=Ubuntu+Condensed|Open+Sans:400italic,700italic,400,700&amp;subset=latin,latin-ext' type='text/css' media='all' /&gt;   ...wins the bounty. How can I correct this code to get an answer?