proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$
Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$
MacBook is displaying the message "no hardware installed" when I click on Wi-Fi I have a MacBook Pro 2009 running MacOS El Capitan. Recently the Wi-Fi has stopped working.   When I click on the Wi-Fi icon it displays the message "No hardware installed", but Ethernet still works.   Is there any way of fixing this by using a Wi-Fi dongle or something of the sort, if so do I need a specific dongle and how do I install it? I run a large bulletin board and I get spammers every now and again. My moderation team does a good job filtering them out but every time I IP ban them they seem to come back (I'm pretty sure it's the same person on some occasions, as the post patterns are exactly the same as are the usernames) but I'm afraid to ban them by IP address every time.  If they are on a dynamic IP address, I could be banning innocent users later down the line when they try to get to my forum through SERPs, but if I ban only via static IPs I know that I'm only banning that one person.  So, is there a way to properly determine if an IP address is static or dynamic?  Thanks. How to type directly with keyboard the U+200C character? How to type directly with keyboard the U+200C character in Notepad or other places... in Windows 8.1? (without depending on a certain font)  I don't want to copy and paste ways. I don't want any complicated ways. I just want something like Alt+### for typing U+200C Directly at any text area in windows 8.1 without depending to a certain font. Just like ALT+129 for U+200B.  (I do not need ALT+129, That's for U+200B. I need U+200C)  the U+200C character name is: ZERO WIDTH NON-JOINER (ZWNJ)  Here is the U200C character info: Will store bought chicken eggs hatch? I have some files/directories on an external hard drive connected using USB to my Windows XP laptop. I am trying to delete those files but every time I try, I get an error "Cannot delete ....: Access Denied. Make sure disk is not full or write protected and that the file is not currently in use".  I checked that the directory is not is use. However, the permissions are set to read-only. Even if I remove the "read-only" checkbox, in Properties->Attributes, it automatically comes back again. How do I fix this? Is it a virus that is automatically changing it back to read-only?  regards,  JP Why is "bicycle" pronounced differently from other obviously related words? What are Alternative's "some" and "many" useful for? Integration of exponential and square root function Show there exists $g:(0,1)\to\mathbb{R}^2$ such that $h\circ g$ constant, $g$ continuously differentiable and $g$ injective Comparing $\pi^e$ and $e^\pi$ without calculating them How can I  compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ? Use of Website Category Landing Pages OK, here's the situation: A reasonably simple website with a primary navigation menu including drop-down list of sublevels on hover. Each navigation level may have 3 - 6 sublevels below it, none more important than the other. For example see this mockup:    My question is: What should happen if the user Clicks on one of the primary navigation options (such as 'News') instead of just hovering over and choosing a sublevel?  My instinct tells me that I should allow a click of this navigation option; it's expected user behavior to be able to click navigation options. If clicked then a landing page of some sort will be present listing all the sublevels (latest news, blog posts, case studies'). However there is no content to add to this landing page other than presenting a list of links, which is pretty useless. I could pad it out by adding iconography and descriptions of the sublevels, but that's just content for the sake of it; does 'Latest News' really need a description in order to understand what it is?  If I don't include a landing page then clicking this primary navigation will have to open something but if I open the 'Latest News' page that risks the user not knowing about the other sections of the site.  (The landing page will be useful for breadcrumb navigation too, so I don't really want to do away with it).  Is my thinking that a landing page showing the list of sublevels actually useful even though there is no actual content to display, or is there a better alternative to go with? Let $A,B,C$ be sets, and $B \cap C=\emptyset$. Show $|A^{B \cup C}|=|A^B \times A^C|$ Let $A,B,C$ be sets, and $B \cap C=\emptyset$. Show $|A^{B \cup C}|=|A^B \times A^C|$ by defining a bijection $f:A^{B \cup C} \rightarrow A^B \times A^C$.  Any hints on this one?  Thank you! What is "%AppData%"? Pre-1972 sci-fi short story or novel: alien(?) tunnel where people try new moves and get destroyed if they're not the correct ones It might be a short story or it might be a novel. It must be before 1972 because I read it in my youth and I am OLD.  The following is very impressionistic.  There is a multidimensional/not of this Earth/alien artifact-like tunnel.   Picture a long, perhaps transparent, storm drain like structure sitting in a field but with an opening on only one end.  It is the subject of a lengthy investigation/exploration that has cost many lives. An instrumented investigator enters the "tunnel" and makes the exact movements that his predecessors have made, advancing until he must do something original at which point he survives and a new "move" is added to the list of correct moves or he is destroyed and a "move" is added to the list of things to be avoided at that point.  I remember absolutely nothing else about the work except that (I hope central) feature. Is the Echo of a Echo Knight actually a creature? I've always been certain that the echo created by an Echo Knight fighter using their 3rd-level feature Manifest Echo is not defined in game terms as a creature, since the description of the echo never explicitly tells it is a creature. But the presence of people believing the echo to be a creature in lieu of the fact that it is immune to all conditions and might make saving throws (game characteristics typically associated with creatures only) brought this doubt to my mind.  from Jeremy Crawford implies it is just an image and not a creature, further confirming that the feature should tell you, but  implies that the echo is an object whilst the feature doesn't explicitly say so. (Included for reference and insight, since tweets are not eligible as official ruling) Is the echo a creature? How to extend the background of an image? Does the Sun produce audible sound? Theoretically if I were able to build some sort of device that let me sit 1 foot away from the surface of the Sun (or any star for that matter) without being vaporized, would a star produce any sort of audible sound at that distance?   From my understanding, sound waves need a medium to propagate through (air, water, etc).   Does this mean that all stars are completely silent (in the audible spectrum of frequencies at least) as they fuse  hydrogen into helium?   I know stars emit radio, xray, infrared, ultraviolet electromagnetic waves, but how about audible (20 Hz to 20Khz) frequencies?   For some reason it seems like this process would produce some sort of sound. If $A=AB-BA$, is $A$ nilpotent? Apache error "Could not reliably determine the server's fully qualified domain name" When I restart my Apache server using the command  sudo /etc/init.d/apache2 restart   I get the following error:  Restarting web server apache2 apache2: Could not reliably determine the server's fully qualified domain name, using 127.0.1.1 for ServerName ... waiting apache2: Could not reliably determine the server's fully qualified domain name, using 127.0.1.1 for ServerName   Is the server using 127.0.1.1 instead of 127.0.0.1? What causes this error? What's the easiest way to unrar a file?