Does the domain of the function $y=\sqrt[3]{x^3+1}$ include $x&lt;-1$? If yes, why is Mathematica and Wolfram Alpha not plotting that part of the function?
Excuse my lack of knowledge and expertise in math,but to me it would came naturally that the cubic root of $-8$ would be $-2$ since $(-2)^3 = -8$.  But when I checked Wolfram Alpha for , real it tells me it doesn't exist.  I came to trust Wolfram Alpha so I thought I'd ask you guys, to explain the sense of that to me.
Showing that a sequence is monotone and thus convergent Here's the original question:     Let $(a_n)$ be bounded. Assume that $a_{n+1} \ge a_{n} - 2^{-n}$. Show   that $(a_n)$ is convergent.   Okay, I know that if I can show that if the sequence is monotone, I can conclude that it is convergent. But I am not sure how to show that it is monotone.  I know that  $$a_n \le a_{n+1} + \frac{1}{2^n} &lt; a_{n+1} + \frac{1}{n}$$  It looks to me as if it is monotonically increasing but I'm quite not sure how to prove my claim. Any hints would be appreciated. Python: using a recursive algorithm as a generator Reconciling different definitions of left-exact functor Version: Ubuntu 16.04  I have used it some weeks, but now it gives me this message when I'm trying to install software and update or upgrade:  Could not get lock /var/lib/apt/lists/lock - open (11: Resource temporarily unavailable) /var/lib/apt/lists/ locked   followed by this:    .  and another error:  ** (appstreamcli:4560): WARNING **: Exception: Unable to get write lock on /var/cache/app-info/xapian/default: already locked   What should I do? Why do some tests have a (nonzero) minimum score? Vandermonde Determinant This is an exercise from Ian Stewart's Galois Theory, $3^{rd}$ edition:     If $z_1,z_2,\ldots,z_n$ are distinct complex numbers, show that the determinant $$D=\left|\begin{array}[cccc]  11&amp;1&amp;\cdots&amp;1\\ z_1&amp;z_2&amp;\cdots&amp;z_n\\ z_1^2&amp;z_2^2&amp;\cdots&amp;z_n^2\\ \vdots&amp;\vdots&amp;\ddots&amp;\vdots\\ z_1^{n-1}&amp;z_2^{n-1}&amp;\cdots&amp;z_n^{n-1} \end{array}\right|$$   is nonzero.       Hint: Consider the $z_j$ as independent indeterminates over $\Bbb C$. Then $D$ is a polynomial in the $z_j$, of total degree $0+1+\cdots+(n-1)=\frac{1}{2}n(n-1)$. Moreover, $D$ vanishes whenever $z_j=z_k$ for $k\neq j$, as it then has two identical rows. Therefore $D$ is divisible by $z_j-z_k$ for all $j\neq k$, hence it is divisible by $\prod_{j&lt;k}(z_j-z_k)$. Now compare degrees.   My question is how to do this. I follow the hint and I'm thinking of $D$ as a cofactor expansion; is that the idea? Also, the fact that $D$ vanishes whenever $z_j=z_k$ for $k\neq j$ follows since it would have two identical columns, right? I don't see why $z_j=z_k$ would imply it has two identical rows, as the hint suggests. The fact that it is divisible by $(z_j-z_k)$ is easy since that would make $z_k$ a root for $k=j$, but I'm not quite sure I follow why the degrees would be different. I think the degree of the product would be $1+2+\cdots+(k-1)$, but then add these up for all $k&lt;n$?   I appreciate any clarification that you can provide! How to detect if a function is called as constructor? Given a function:  function x(arg) { return 30; }   You can call it two ways:  result = x(4); result = new x(4);   The first returns 30, the second returns an object.  How can you detect which way the function was called inside the function itself?  Whatever your solution is, it must work with the following invocation as well:  var Z = new x();  Z.lolol = x;  Z.lolol();   All the solutions currently think the Z.lolol() is calling it as a constructor. If you have the ability to cast a spell without a spell slot, can you cast that spell using a spell slot? A 10th level Drow warlock (for instance) has the following Drow Magic racial trait (PHB p. 24), in addition to its warlock spellcasting abilities:  You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.  Can the Drow warlock cast faerie fire (which is not on the warlock spell list) using a warlock spell slot? Can it cast darkness (which is on the warlock spell list) using a warlock spell slot?  The  says this about the Magic Initiate feat:  If you’re a spellcaster, can you pick your own class when you gain the Magic Initiate feat? Yes, the feat doesn’t say you can’t. For example, if you’re a wizard and gain the Magic Initiate feat, you can choose wizard and thereby learn two more wizard cantrips and another 1st-level wizard spell. If you have spell slots, can you use them to cast the 1st-level spell you learn with the Magic Initiate feat? Yes, but only if the class you pick for the feat is one of your classes. For example, if you pick sorcerer and you are a sorcerer, the Spellcasting feature for that class tells you that you can use your spell slots to cast the sorcerer spells you know, so you can use your spell slots to cast the 1st-level sorcerer spell you learn from Magic Initiate. Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.  But this specifically requires you to pick the feat for one of your classes to be able to cast the spell learnt using your class spell slots, and does not seem directly applicable. A Warlock can use warlock Pact Magic spell slots to cast spells learnt from levels of another spellcasting class, per the multiclass spellcasting rules (PHB p. 164):  If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.  Similarly, if the warlock has learnt the invocation Mask of Many Faces (PHB p. 112)...  You can cast disguise self at will, without expending a spell slot.  ...Can the warlock cast disguise self using a spell slot, e.g. to decrease the chance it can be dispelled? Export/Deploy Drupal site/page to static? (no cache) I wonder if there is an existing  module that can help in the subject. I am not looking for a caching module (like Boost or Varnish), but a way to export my whole site (or only certain pages) into static.  The idea is that I want my web server will be a very simple and light and will only serve static files (html/css/js) - no database on production, no php engine will be installed on server. All the Drupal engine will be on another server, export will be done each time adding/editing/changing a node or view, and a new set of html files will be created in a directory structure.  These files will later on be copied to the production site that will serve always and only static files, no .htaccess that will check requests if file exist etc.   Is there a module for that, or a workaround that this can be achieved with Drupal?  Thanks Stop jQuery .load response from being cached I have the following code making a GET request on a URL:  $('#searchButton').click(function() {     $('#inquiry').load('/portal/?f=searchBilling&amp;pid=' + $('#query').val());             });   But the returned result is not always reflected. For example, I made a change in the response that spit out a stack trace but the stack trace did not appear when I clicked on the search button. I looked at the underlying PHP code that controls the ajax response and it had the correct code and visiting the page directly showed the correct result but the output returned by .load was old.  If I close the browser and reopen it it works once and then starts to return the stale information. Can I control this by jQuery or do I need to have my PHP script output headers to control caching? What is Interface method in salesforce Generate mesh at run time Is it possible to generate mesh at runtime in blender game engine? That is, is it possible for blender to generate meshes using python code when the game runs?  I really would like to know if this is possible, as my final year project requires use of this. Trouble understanding the Bohr model of the atom Word for "blissfully oblivious"? What overwrites /etc/resolv.conf on every boot? I was given the files for a mini linux, that boots directly into firefox. It works for all it should be doing, only that I do not get an internet connection.  We have 3 DNS servers in the network, which all work. I can ping them, too. But when trying to ping google.de or wget google.de I get a bad address error.  nslookup google.de works for some reason.  I tracked the issue down to my resolv.conf on the booted system not having the same contents as the resolv.conf that I put into the .iso file.  I tried understanding all the factors that go into creating and modifying resolv.conf. I'm not quite sure I got it all, but I definitely didn't find my solution there.  So as a last ditch effort, I tried making the resolv.conf file immutable using   :~# chattr +i /etc/resolv.conf   When rebuilding and booting again to my surprise my file was renamed to resolv.conf~ and in its place was the same standard file that has been haunting me.   The file contents make me believe it gets the information from the network itself. When starting the .iso in Virtualbox without internet access, my file is being kept as it is.  I tried changing /etc/dhcp/dhclient.conf to not get the information from the net, by deleting domain-name-server and domain-name-search from the request part of the file.  Didn't work unfortunately.  I don't have the NetworkManager installed. The iso is based on Ubuntu 14.04.  There is probably vital information missing. I'm happy to provide it.  UPDATE:  I think I found the file that clears resolv.conf.  It seems to be /usr/share/udhcpc/default.script  #!/bin/sh  # udhcpc script edited by Tim Riker &lt;Tim@Rikers.org&gt;  [ -z "$1" ] &amp;&amp; echo "Error: should be called from udhcpc" &amp;&amp; exit 1  RESOLV_CONF="/etc/resolv.conf" [ - n "$broadcast" ] &amp;&amp; BROADCAST="broadcast $broadcast" [ -n "$subnet" ] &amp;&amp; NETMASK="netmask $subnet"  case "$1" in     deconfig)         /bin/ifconfig $interface 0.0.0.0         for i in /etc/ipdown.d/*; do             [ -e $i ] &amp;&amp; . $i $interface         done         ;;      renew|bound)         /bin/ifconfig $interface $ip $BROADCAST $NETMASK          if [ -n "$router" ] ; then             echo "deleting routers"             while route del default gw 0.0.0.0 dev $interface ; do                 :             done              metric=0             for i in $router ; do                 route add default gw $i dev $interface metric $((metric++))             done         fi          echo -n &gt; $RESOLV_CONF # Start ----------------           [ -n "$domain" ] &amp;&amp; echo search $domain &gt;&gt; $RESOLV_CONF         for i in $dns ; do             echo adding dns $i             echo nameserver $i &gt;&gt; $RESOLV_CONF         done         for i in /etc/ipup.d/*; do             [ -e $i ] &amp;&amp; . $i $interface $ip $dns          done # End ------------------         ;; esac  exit 0   It's part of the udhcpc program. A tiny dhcp client, that is part of busybox  Will investigate further.    UPDATE2 AND SOLUTION:  I commented the part out (#Start to #End), that seemingly overwrites the /etc/resolv.conf file and sure enough. That was the culprit. So an obscure script caused all this trouble.  I changed the question to reflect, what actually needed to be known to solve my problem, so it would be easier to find for people with the same problem and so I could accept an answer.  Thanks for the help here in figuring things out. How to run find -exec? What is the term for the grandson of my aunt? Comment on Duplicate Reference: Why would this be marked duplicate when it was asked years prior to the question referenced as a duplicate?  I also believe the question, detail, and response is much better than the referenced question.  I've been a C++ programmer for quite a while but I'm new to Java and new to Eclipse.  I want to use the  to visualize some data I'm working with.  This code is organized like this:  ./com ./com/touchgraph ./com/touchgraph/graphlayout ./com/touchgraph/graphlayout/Edge.java ./com/touchgraph/graphlayout/GLPanel.java ./com/touchgraph/graphlayout/graphelements ./com/touchgraph/graphlayout/graphelements/GESUtils.java ./com/touchgraph/graphlayout/graphelements/GraphEltSet.java ./com/touchgraph/graphlayout/graphelements/ImmutableGraphEltSet.java ./com/touchgraph/graphlayout/graphelements/Locality.java ./com/touchgraph/graphlayout/graphelements/TGForEachEdge.java ./com/touchgraph/graphlayout/graphelements/TGForEachNode.java ./com/touchgraph/graphlayout/graphelements/TGForEachNodePair.java ./com/touchgraph/graphlayout/graphelements/TGNodeQueue.java ./com/touchgraph/graphlayout/graphelements/VisibleLocality.java ./com/touchgraph/graphlayout/GraphLayoutApplet.java ./com/touchgraph/graphlayout/GraphListener.java ./com/touchgraph/graphlayout/interaction ./com/touchgraph/graphlayout/interaction/DragAddUI.java ./com/touchgraph/graphlayout/interaction/DragMultiselectUI.java ./com/touchgraph/graphlayout/interaction/DragNodeUI.java ./com/touchgraph/graphlayout/interaction/GLEditUI.java ./com/touchgraph/graphlayout/interaction/GLNavigateUI.java ./com/touchgraph/graphlayout/interaction/HVRotateDragUI.java ./com/touchgraph/graphlayout/interaction/HVScroll.java ./com/touchgraph/graphlayout/interaction/HyperScroll.java ./com/touchgraph/graphlayout/interaction/LocalityScroll.java ./com/touchgraph/graphlayout/interaction/RotateScroll.java ./com/touchgraph/graphlayout/interaction/TGAbstractClickUI.java ./com/touchgraph/graphlayout/interaction/TGAbstractDragUI.java ./com/touchgraph/graphlayout/interaction/TGAbstractMouseMotionUI.java ./com/touchgraph/graphlayout/interaction/TGAbstractMousePausedUI.java ./com/touchgraph/graphlayout/interaction/TGSelfDeactivatingUI.java ./com/touchgraph/graphlayout/interaction/TGUIManager.java ./com/touchgraph/graphlayout/interaction/TGUserInterface.java ./com/touchgraph/graphlayout/interaction/ZoomScroll.java ./com/touchgraph/graphlayout/LocalityUtils.java ./com/touchgraph/graphlayout/Node.java ./com/touchgraph/graphlayout/TGAbstractLens.java ./com/touchgraph/graphlayout/TGException.java ./com/touchgraph/graphlayout/TGLayout.java ./com/touchgraph/graphlayout/TGLensSet.java ./com/touchgraph/graphlayout/TGPaintListener.java ./com/touchgraph/graphlayout/TGPanel.java ./com/touchgraph/graphlayout/TGPoint2D.java ./com/touchgraph/graphlayout/TGScrollPane.java ./TG-APACHE-LICENSE.txt ./TGGL ReleaseNotes.txt ./TGGraphLayout.html ./TGGraphLayout.jar   How do I add this project in Eclipse and get it compiling and running quickly? I'm on Ubuntu 16.04.1 at the moment, and system windows like the Files window, Terminal or System Software open behind the Unity launcher, which means they are partially hidden behind it. I can move them with Alt+F7, but I'd like a more permanent solution to this problem. Show that there is no epimorphism from ($\mathbb{Q}$, +) to ($\mathbb{Z}$, +)