I read in a paper on set theory that the supremum and the infimum of the empty set are defined as $\sup(\{\})=-\infty$ and $\inf(\{\})=\infty$. But intuitively I can't figure out why that is the case. Is there a reason why the supremum and infimum of the empty set are defined this way?
Let $E$ be an empty set. Then, $\sup(E) = -\infty$ and $\inf(E)=+\infty$. I thought it is only meaningful to talk about $\inf(E)$ and $\sup(E)$ if $E$ is non-empty and bounded?   Thank you.
This winter I am planning to track my downhill skiing/snowboarding using a GPS.  Most of my riding will occur at the same resort.  I would like to be able to create a sort of "heatmap" that shows the amount of runs that I have made in a given area.  As I add more and more GPS traces to my database, my goal would be to see a sort of linear heatmap of the most traveled areas.  Given the nature of downhill skiing, you would expect the uphill chairlift lines will be the "hottest" because it will be the only places visited over-and-over again.  Given that 1) my track will not be the same every time and 2) the area covered by following one "run" may be a few hundred feet wide, what might be the best way to analyze this "linear" data to create a sort of heatmap?  My thoughts were to buffer the lines, then intersect the polys to get a sort of Venn Diagram thing going.  My preference is to use open-source technologies.  I've got QGIS and PostGIS loaded and available.  UPDATE: In regards to @blah238's response, I was thinking of something that might be able to "collect" the number of passes ("runs") through an area, and then symbolize by the count.  Conceptually, this would be similar to ArcGIS "" (but for lines, not points) or  (but for multiple line in roughly the same area).  A more visual example of a similar concept might be a traffic-flow map, where highly-congested areas would equate to "highly-traveled" ski runs/areas:    I've read the following questions that may give some ideas, but they don't really address what I am trying to accomplish: Logo Pack - What should I include? What files should i put in a logo pack to give to clients? Should I put small, medium and large version of my logo as well as the vector file. How to select records with maximum values in two columns? A challenge on the parabola Good Night ... Here I am editing the question and explaining the purpose of the same ... Mathematics course, and I love this science, I am very curious, and these days, studying a book of calculus found in (book from the library of the University) a sheet full of challenges, to share with a colleague, he gave me this challenge, which he also failed to respond ... I tried, I tried, I used the formulas of quadratic equation, and I've been trying since Friday (here in Brazil is Sunday at 22:33), and could not, and I ask your help, and please believe me, I struggled, tried to make substitutions, tried both, and I'm confused ... I must find the values of $ a, b $ and $ c $?  Consider a parabola whose roots, other are $x'$ and $x''$, and such that $x'=-x''$ where $x'&gt; 0$. Let $$A = (x', 0)$$ $$B = (x'', 0)$$ and $V$ the vertex of the parabola. Suppose that the equation of this parabola is $$y = ax^2 +bx+ c$$. Calculate the value of the discriminant $$\Delta=b^2-4ac$$, so that the triangle is equilateral AVB. Most of the time, it should be really easy to determine who is my ally and who is not. The people who try to kill me are most likely my foes, whereas the people who entered the dungeon with me are probably my allies.   But sometimes it’s not that simple. What if one of my party members got replaced by an evil doppelganger, who plans to kill me in the near future? What if one of my allies infiltrated the court of the evil king and I accidentally have to fight her, while she is (unbeknownst to me) under the effect of a  spell? What if a character fell in love with the BBEG and just wants to save him from himself, while the BBEG hates the character and actively tries to murder him?  Most spells and abilities that affect others specify the targets as “creatures” or “willing creatures”. Some explicitly affect allies only. That’s why I want to know, how to determine who’s allied with whom. Do I have to think that someone is my ally for someone to count as my ally? Does someone else have to believe to be my ally in order to count? Do both of us have to agree on being allies? Is there an objective observer who determines who’s allied? Can Person A be the ally of Person B, while Person B is not the ally of Person A?  There are many examples for spells/abilities that care about allies, among others:   the Exalted Champion feature from the  (SCAG, p. 132-133), that grants allies within 30 feet advantage on Wisdom saving throws. the Inspiring Surge feature from the  (SCAG, p. 128), that allows one or two allies within 60 feet to make a melee weapon attack as a reaction.   With features like this, you could accidentally affect someone who’s not actually your ally. You could fail to affect someone who is actually your ally, but you don’t know it (yet). Or you could affect someone who doesn’t want to be affected (because they don’t want to be allied with you). This all depends on the interpretation of the word “ally”. I’m not sure if "ally" is a defined game term (but I don’t think so), or if I have to rely on the dictionary definition. That’s why I want to ask:  How do I determine who’s allied with whom? How can I modify the advanced search form when my hook runs too early? I'm using  to modify the content of the form on the search page &mdash; I want to remove the "phrase" and "not" fields, and restrict the "only of the types" checkboxes to a hand selected list of types.  I have the following code, which is being called and which I think should work:  function MYMODULE_form_search_form_alter(&amp;$form, &amp;$form_state, $form_id) {     unset($form['basic']);     unset($form['advanced']['keywords']['phrase']);     unset($form['advanced']['keywords']['negative']);     unset($form['advanced']['type']);     $form['advanced']['type'] = array(         '#type' =&gt; 'checkboxes',         '#title' =&gt; t('Only of the type(s)'),         '#options' =&gt; array(             'blog' =&gt; 'Blog entry',             'event' =&gt; 'Event',             'news' =&gt; 'News article'         )     );     // print_r($form);     // die; }   Except  gets to the search form after my form alter hook runs, so I don't get a chance to modify any of the form elements that it's adding.  I'm running Drupal 7.22.  How can I modify the advanced search form when my module hook runs before the hook that defines the advanced search form? Examples where $H\ne \mathrm{Aut}(E/E^H)$ How do you run a 32-bit program on a 64-bit version of Ubuntu? I have a 64-bit version of Ubuntu 13.10.  I want to run a 32-bit version of a program.  However I see ia32-libs is not included in 13.10.  Is there a replacement package for this or what? 0 ram free on ICS I'm new to the ICS os, coming from Gingerbread from another phone. The phone's quite nice and fast when I first used it, but just this morning it started exhibiting weird actions. Camera360 doesn't open up automatically when I click it, the Camera closes for no reason, Pocket restarts when an article has an image, and TTPod resets the song to the previous song on my widget in the middle of a new song I'm playing.  I checked out the ram, and and was surprised to see 500+ mb used, 0 mb available. I looked at the list of programs and I see the usual culprits like Facebook, but the ram they're using doesn't add up to 500+ mb. Yeah, I know that the whole ram indicated isn't usable by the apps, but I don't think I've done anything to warrant the 0 mb. I close my apps. There are times when there's free 4 to 5 mb, but I think that too shouldn't be the case. I tried installing Advance Task Killer, it listed less than 10 apps, killed it, still the ram is what it is above.   I did a reboot, and now it reads 300+ mb usage, 200 something free.   What can I do? How do I ensure an app's closed on ICS? I installed GoLaucher because it has a handy task killer and I can see the ram freed in an instant, but I would like to know if there's a better way to handle the RAM. Wouldn't multiple filters in a convolutional layer learn the same parameter during training? Based from what I have learned, we use multiple filters in a Conv Layer of a CNN to learn different feature detectors. But since these filters are applied similarly (i.e. slided and multiplied to regions of the input), wouldn't they just learn the same parameters during training? Hence the use of multiple filters would be redundant? How to install Anaconda on Ubuntu? Using "that" and "this" interchangeably Learning and using English I'm always confused about what word to use for referring to things that have been described by me a few sentences earlier: "that" or "this".   Confusion comes from the fact that only the equivalent of "this" is always used in my native language for such referring. But I've noticed that in English for such referring "that" is used as often as "this" (or maybe even more often).  Some examples for illustrating (just tried to google something appropriate to convey my idea better):     We assess local... the demographics of the local population. What are the natural traffic drivers in the area. Things like cinemas and pubs and retail and office and all that type of things and we now put together a bit of a matrix and actually give a weighted score to each of the things we know help our business. That helps us decide in a more scientific fashion.   or:     The problem of Cervantes' origin became after that into a tough matter. Some experts believed that the Cervantes from Alcázar, in the times of the Lepanto Battle, was in the age of a child, more concerned about gathering nests and that type of things than about fighting as a soldier.   and for "this":     I believe lot of people who involved fishery industries in Mexico Gulf are suffering now but as for economic issues, U.S government and other countries will support them and give an utmost response to it. Important thing is that we learn from this mistake and make sure this type of things will never happen in the future and protect nature environment thus we have to take this technology to get energy from water very seriously. I really hope this technology will be available and used for everybody as soon as possible.   Is there a rule describing proper usage of these words in cases like this? (or should I have written "..in cases like that"?) Can we get our personal voting data displayed on the Questions lists? There's been several times where I have been surfing the site, and noticed a question that I am not sure I've checked. Even if I've checked the question before, I can't recall off-hand whether or not I've voted on it or its answer.  Is there a way we can have our personal voting history reflected to some degree or another, in the Questions listing? For example, could there be an indicator as to whether or not we have already voted (up or down) on a question, and perhaps maybe an additional indicator as to whether we've up-voted any of its answers?  It would be very convenient to have some of this data available up-front, without actually having to view the question again. How does Meta Stack Exchange work? Meta Stack Exchange seems to have different rules and user behavior than the "regular" Stack Exchange sites.  How is it different?    Individual questions answered below:                Related I get the following error when i do sudo apt-get update  W: GPG error: http://ppa.launchpad.net precise Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 6AF0E940624A220   I tried following ways after reading various questions here and also by googling.  I tried  sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys 6AF0E1940624A220   It did not help. I also tried  sudo gpg --keyserver hkp://keyserver.ubuntu.com:80 --recv-keys   thinking port 11371(default) may be blocked due to firewall.  I also tried to reset everything as follows  sudo rm /var/lib/apt/lists/* -vf sudo apt-get update &amp;&amp; sudo apt-get upgrade   still i get the same error or get update. Any other possible solution I can try? Configuring Firefox to use a proxy from the command line I'm looking for a way to tell Firefox to use a proxy in Ubuntu Ludid (10.04.1 LTS).   I'm installing Ubuntu semi-automatically on several identical machines. For this, I use a script that, amongst other things, sets system-wide proxy settings:  echo "export http_proxy=http://myproxy:3128/" &gt; /etc/profile.d/proxy.sh echo "export HTTP_PROXY=http://myproxy:3128/" &gt;&gt; /etc/profile.d/proxy.sh chmod a+x /etc/profile.d/proxy.sh   I can confirm that the variables are set correctly when checking in a shell. Most programs respect this setting and use my proxy. Firefox, however, does not. When I open its network connection settings dialog, however, it is pre-set to "use system settings" for the proxy.   What does Firefox expect here? A differently-named variable? Can I set something in FF's prefs.js? extract string between double quotes I have requirement that i need to extract string using quotes. My string as follows.  "abcd efgh" "ijkl mnop" "qrst uvwxyz"   can you help me to get the string between second double quotes(ijkl mnop) using sed or grep command.  In other words if I say give me the string in first quotes I want the  first string; if I say second string it should give me string between second double quotes and similarly third one also. Why does inductor current lag the applied voltage at its terminal by 90 degrees? I studied electromagnetism and I am currently working with Inductors.  I could not figure out the physical reason based on electromagnetics on why inductor current lags the applied voltage by 90 degrees?  Everywhere I find the Formula as the starting point of explanation which is partially satisfying.    Induced EMF =L di(t)/dt    From the above equation, I understood that when there is a change in current there will EMF induced in the coil terminals. But How this induced EMF makes the current to lag the supply voltage V(t) = Vm*sin(wt)? (Please don't just differentiate and say there is a lag, I am looking for a physical sense on why this happens)  Lets us consider the AC voltage is applied across the terminal at time t=0; At this time there is almost no current. From this transient state how the steady-state is reached where the current is delayed with respect to the voltage?  Can someone help me to figure out why the current lags the Supplied voltage by using electromagnetism, the motion of Charges or any other basic concepts? What is Elrond's position/status, as compared to Galadriel's? In the Lord of the Rings books, Galadriel of Lorien is even more exalted than in the movies.  She is essentially a queen, and the deference shown to her far exceeds that shown to Elrond of Rivendell.  Yet in the films, we see her trying to persuade Elrond to intervene in the battle of Helm's Deep by sending Elf troops to reinforce Theoden's forces.  Eventually, Elrond relents, and orders the Elves into battle.  But the troops in question are Galadriel's, not Elrond's - they are led by Haldir, one of Galadriel's guards; the trip from Rivendell to Helm's Deep (500 miles as the crow flies) is about 2 times longer than the trip from Lorien to Helm's Deep (250 miles as the crow flies), far too long for the reinforcements to arrive in time.  But then Haldir greets Theoden of Rohan by saying he has come bearing a message from Elrond - not Galadriel.     I bring word from Lord Elrond of Rivendell. An Alliance once existed between Elves and Men. Long ago we fought and died together. We come to honor that allegiance.    This seems odd for a few reasons. The troops are Galadriel's, not Elrond's, and they come from Lorien, not Rivendell. She clearly wanted to send them to Helm's Deep - she was cajoling, prodding, and guilt-tripping him into doing just that.  She is also a queen, and could presumably order them into battle regardless of what Elrond says.    So why does she have to talk Elrond into letting her send her own troops into battle, especially when he is too far away to contribute manpower (elfpower) or materiel to the battle of Helm's Deep?  Why not say, "I'm not going to sit here doing nothing while Saruman runs roughshod over Rohan;  you can support me or not, Elrond, but if you aren't willing to help, you can at least get the hell out of the way"?  Why does her aide Haldir "bring word from Elrond" when the troops - and the impetus for sending them - came from Galadriel, and Elrond resisted sending them out for as long as he could?  And again, what is Elrond's official position/status/rank, and why would a queen need his permission to deploy her own troops? What happens to a Devil when its alignment is forcibly changed? According to PHB 122, in the section entitled Alignment in the Multiverse, for most creatures, "alignment is a moral choice." Myth says good-aligned gods created the races that are free to choose to be evil (they have free will), while evil-aligned gods created the races that are very inclined towards evil, since they were born to serve the evil god in the first place.  However, there are some types of creatures -- like devils -- which embody an alignment in its essence:     Alignment in the Multiverse, PHB 122      A devil does not choose to be lawful evil, and it doesn't tend toward lawful evil, but rather it is lawful evil in its essence. If it somehow ceased to be lawful evil, it would cease to be a devil.   My first thought was to not take this reading literally. The text could have meant that a neutral evil devil "isn't really a devil," but really, it's still a devil. It's a bit like imagining a guitarist who loses both hands. The loss of some part of a person's essence makes them not the same person anymore.  However, keeping my attention on it, I'm finding this text more and more difficult to interpret. If a devil "is lawful evil in its essence" then what does it mean when "it would cease to be a devil"? What happens to a devil when its alignment is forcibly changed?  There are , and a devil can be subject to some of them.