Change the Windows key to Caps Lock in Windows 7
How can I remap a keyboard key?
On the meta home page I noticed the  is listed as doing something, but cannot figure out what it is.  There is nothing on the  or its , but the recent activity is listed with the same timestamp. Prove that the union of the interior of a set and the boundary of the set is the closure of the set I'll denote closure of A with $\overline{A}$, $A^\circ$ as the interior of A, $\partial A$ as the boundary of $A$ and $A'$ as the accumulation points. I'm trying to prove the following:   $$\overline{A}=A^\circ \cup\partial A$$  Is this a valid proof?  Take $x \in A^\circ \cup \partial A$ then $x \in A^\circ$ or $x \in \partial A$   if $x \in A^\circ$ then $x \in \overline{A}$  if $x \in \partial A$ then $x \in \overline{A} \cap\overline{(X\setminus A)}$ thus $x \in\overline{A} $ so $A^\circ\cup\partial A\subset\overline{A}$  Take $x \in \overline{A}$ then $x \in A' \cup A$ thus $x \in A'\setminus A$ or $x \in A^\circ$  if $x \in A'\setminus A$ then $x \in \overline{(X\setminus A)}$ so $x \in \overline{A}\cap\overline{(X\setminus A)}$ and $x \in\partial A$ so $x\in A^\circ\cup\partial A$  if $x \in A^\circ$ then $x \in A^\circ\cup \partial A$ so $\overline{A}\subset A^\circ\cup\partial A$ Is mass converted into energy in exothermic chemical/nuclear reactions?  My (A Level) knowledge of chemistry suggests that this isn't the case. In a simple burning reaction, e.g.   $$\mathrm{C + O_2 \to CO_2}$$  energy is released by the $\mathrm{C-O}$ bonds forming; the atoms lose potential energy when they pull themselves towards each other, in the same way that a falling object converts GPE to KE. There are the same number of protons, electrons etc. in both the reactants and products. I would have assumed that this reasoning extends to nuclear fission/fusion as well, but one physics textbook repeatedly references very small amounts of mass being converted into energy in nuclear reactions.  So I just wanted to know if I was wrong about either of these types of reactions, and if so, what mass is lost exactly? "Temporarily unreachable" when using fetch as Googlebot from Webmasters Tools The page I'm trying to fetch is public.  The server logs show no request.  Google reports that any page on my site is "Temporarily unreachable". It won't even fetch the verification file, a static HTML file that was successfully used to verify with the webmaster tool application itself.   My site is running on the .  What is causing this? Complete references in between the document When we do   \bibliographystyle{style} \bibliography{References.bib}   We get all the references at the end of the document, with some kind of reference/id in the document.   What I need is the complete reference in the document itself (not at the end) and then to be able to write what I want below this reference. For example;  Chaminade, T., Hodgins, J., &amp; Kawato, M. (2007). "Anthropomorphism influences perception of computer-animated characters' actions." Social Cognitive and Affective Neuroscience 2(3): 206-216      - Some text..........      - Some more text.......  Cowley, S. J., Kanda, T. (2005). "Friendly machines: interaction-oriented robots today and tomorrow." Alternation 12(1a): 79-106.      - Some text.....      - Some more text....   How can I do it? Thanks. Custom question lists April status update Previous posts:                           After a long hiatus, a new Custom Question Lists update is out with a fresh new design.  What's shipped?  If you are participating in the Custom Question List alpha (by ), you'll see some brand new UI components when viewing question lists:  New tab bar and editor    Based on your feedback, we decided to rethink the whole UI, removing some confusing and redundant components, and bringing back some much loved pieces.  The new design is closer to "the existing question lists with advanced options" than "a radically new experience".  Huge shoutouts to our new designer, , for making this new design possible and to  for picking up development while I was on sabbatical!  Some important notes for existing testers:   The summary is back in its rightful place next to the tab bar.  If bounty or unanswered filters applied, the summary provides a clearer explanation of what you're looking at, inspired by /unanswered. The "Filter" and "Sort" popovers are gone. Sort was too similar to the existing tabs, and Filter was a form in a popover. (how were you supposed to submit or cancel?)  Now there's a dedicated UI for filtering. The complex "Create" and "Edit" modals are gone.  Now you just configure your list with the same UI as if you were filtering and click "Save filter" when you're ready to save. You can preview your changes before updating a filter.  Editing a saved filter and clicking apply shows you your new options while remembering what filter you were looking at.  You can then click "Save filter" to either update your existing filter or create a new one. We handle responsive better.  The tab bar has been set up so it compresses more naturally on narrower browsers and mobile devices.   New sidebar widget    A new right-sidebar widget has been added, similar to Watched Tags, to give you more direct access to your saved lists.  This is where indicators will live in Phase 3 (which I swear we are actually going to ship).  Please give it a try.  If you have any questions, bugs, or feedback, you can reply to this post directly or add a new question with the  tag.  A wider alpha  Since we're nearing launch, we're expanding the alpha out to the whole network.  So feel free to try this out on any site you frequent, to help suss out any site-specific issues.  What's next?  As always, Phase 3 of this project is adding indicators for new content, now in the right sidebar widget.  There's increasing interest in this feature from Enterprise, so I do really expect this to ship and get out of Alpha sooner rather than later and this time, I mean it. Why Faddeev-Popov ghost cannot exist in external line? In-universe, a lightsaber is basically a controlled, fixed-length plasma stream that can be used as a sword. As such, the blade wouldn't have any mass (or infinitesimal mass if you want to get technical about E=mc2).  Anyone familiar with fencing or swordplay knows that the weight of the handle and blade, and thus the overall balance of the weapon, is critically important while fighting. A sword that is too blade-heavy is unwieldy; it is difficult to start, stop, and change the path of the blade as it's swung, which is vital to outmaneuvering your opponent in a duel. Conversely, a sword with too light a blade (equivalently, too heavy a hilt) is too easy to swing around. it's hard to "feel" where the blade is, and thus to make the minute changes needed to block a blow or land one. It can also be just as exhausting, in a different way, to wield such a sword; because most of the mass is behind your hand, it takes muscle power to point the blade forward instead of upward, while a balanced weapon would point directly forward from the user's hands. You also get no mechanical advantage while swinging it.  Obviously, out-of-universe the props were full "swords" that were then rotoscoped to paint the beam blade over the white wooden (later plastic) dowel that protruded from the saber. This gives the actual prop used by the actors a more realistic weight and balance, which can be seen in the duels (Darth Maul has to heave a bit to get that double-saber staff moving, and you can see the effort behind a full baseball swing of the sword by Luke or Vader in the original trilogy).  Was there any attempt in-universe to explain the weight and balance that an active lightsaber seemed to possess? I have this MySQL query:  SELECT DAYOFYEAR(`date`)  AS d, COUNT(*)  FROM  `orders`  WHERE  `hasPaid` &gt; 0 GROUP  BY d ORDER  BY d   Which returns something like this:  d  | COUNT(*) | 20 |  5       | 21 |  7       | 22 | 12       | 23 |  4       |   What I'd really like is another column on the end to show the running total:  d  | COUNT(*) | ??? | 20 |  5       |   5 | 21 |  7       |  12 | 22 | 12       |  24 | 23 |  4       |  28 |   Is this possible? Convergence of $\int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx$ I need to show that $$ \int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx $$ converges if and only if $\alpha + 1 &lt; \beta$. I used substitution and integration to get $$ \lim\limits_{t\to\infty}\int\limits_1^tx^\alpha\sin(x^\beta)dx \overset{u = x^\beta}{\underset{du = \beta x^{\beta-1}}{=} }\lim\limits_{t\to\infty}\left(\frac{1}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1 - \beta}{\beta}}\sin{u}du\right) $$ If $\alpha + 1 -\beta &lt; 0$, then we have a product if a function with bounded anti-derivative and a differentiable decreasing function which tend to 0, and from Dirichlet's criterion, the integral converges.   For the converse, I used integration by parts $$  = \frac{1}{\beta}\lim\limits_{t\to\infty}\left(\left.-u^{\frac{\alpha + 1-\beta}{\beta}}\cos{u}\right|_1^{t^\beta} + \frac{\alpha+1-\beta}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1-2\beta}{\beta}}\cos{u}du\right) $$ For $\alpha + 1 -\beta\geq 0$, I get that the first term indeed diverges, but this is not enough to show total divergence. How can one show it? Is it possible to know who voted my question/answer up or down? If not, then is it possible to include this feature? By this feature we can come to know if you are being voted up by question owner or persons giving other answers to same question or just the visitors of that question thread.  Like...  If it is a group of people giving other answers, then your answer is most correct If it is readers of that question, then your answer is more popular I came across the following problem in a book:  Give a combinatorial proof of  $$ {n \choose 0} + {n \choose 2} + {n \choose 4} + \, \, ... \, = {n \choose 1} + {n \choose 3} + {n \choose 5} + \, \, ...$$ using the "weirdo" method (i.e., where one of the elements is chosen as special and included-excluded -- I'm sure you get the idea).  After days of repeated effort, the proof has failed to strike me. Because every time one of the elements is excluded, the term would be ${n-1 \choose k}$ and not $ {n \choose k}$, which is not the case in either of the sides of the equation. The problem:  Consider the smooth quadric $Q=V(X_{0}X_{1}+X_{2}X_{3}+X_{4}^{2})\subset\mathbb{P}^{4}$ and the line $L=V(X_{0},X_{2},X_{4})$ contained in it. Prove that there exists a vector bundle $F$ on $Q$ with a section vanishing exactly at $L$.  The hardest thing is to get the transition matrices. Any help is appreciated. Zariski Tangent Space and $k[\varepsilon]/\varepsilon^2$ Let $X$ be a scheme over a field $k$, and let $x\in X$ be a rational point, that is, we have $k(x):=\mathcal{O}_x/\mathfrak{m}_x\cong k$.  Let $\alpha:\mathfrak{m}_x/\mathfrak{m}_x^2\rightarrow k$ be an element of $T_x=(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$, the Zariski tangent space to $X$ at $x$.  Using $x$ and $\alpha$, I'd like to define a $k$-morphism of schemes Spec $k[\varepsilon]/\varepsilon^2\rightarrow X$.  In the course of the construction, I need to define a local map of local rings $\mathcal{O}_x\rightarrow k[\varepsilon]/\varepsilon^2$.  Using $\alpha$, what is a natural way to define such a map?  It might be helpful to note that I have an injection $k\hookrightarrow\mathcal{O}_x$ arising from the $k$-scheme structure on $X$.  I think this leads to a decomposition $\mathcal{O}_x/\mathfrak{m}_x^2=k\oplus(\mathfrak{m}_x/\mathfrak{m}_x^2)$ as $k$ vector spaces, from which the desired map would follow, but I can't seem to explicitly see what this decomposition is inside of $\mathcal{O}_x/\mathfrak{m}_x^2$.  This is part of exercise II.2.8 in Hartshorne. Is it possible to take an image from the clipboard and output it to a file (using X)?  I can do this with text easily:  $ xclip -selection c -o &gt; file.text   But when I try the above with an image nothing is written.    The reason I want to do this is I don't have an image editor installed, and it got me thinking whether I could do this without installing one. Must an ideal contain the kernel for its image to be an ideal? Hard disk file system is shown as Raw file system; how can I access it? We just got a new laptop (ASUS) though we did have a mac.  We are trying to log onto Mojang to download Minecraft and it keeps saying, "The username and password are not recognized."   We have done all the usual procedures; we have reset the password, we are using the email, not the username to sign in, and it looks like the minecraft login servers are working etc. etc.  We quit Minecraft on the old computer and shut off that computer thinking that was the issue.   Still, we can't get onto Mojang.  We are accessing mincraft through firefox and trying to download it to the new computer but we can't get into the account to do that!  What in the world could be the issue? After migration to MacOSX 10.11 "El Capitan", how to run simple utilities such as "texdoc" and "kpsewhich"? Last night, I migrated my MacBook's operating system from 10.10 "Yosemite" to 10.11 "El Capitan". Everything seemed to go smoothly. Happily, regarding my TeX-related work, I am able to use the GUI front ends "TeXworks" and "TeXshop" just fine since completing the OS upgrade. I use MacTeX2015, with all the latest updates installed.  Oddly, though, since completing the upgrade I can no longer run any MacTeX command-line utilities such as texdoc and kpsewhich. (Actually, as I just discovered, I can no longer run pdflatex either from a command line. However, I never do run pdflatex from a command line prompt, so that's not an urgent problem for me -- at least not for now.)  The suggestions given in the posting  do not seem to be applicable to this case, as they all seem to revolve around setting up /usr/texbin correctly -- /usr/texbin is no longer allowed under MacOSX 10.11. The document  by Herb Schulz deals in depth with adjusting settings in the GUI front ends, but it appears not to touch on what to do about /usr/texbin, which appears to be the now-missing link to TeX's command line utilities.   Pointers and solutions most welcome! Having the derivative be an operator