Show that these 5 lines are concurrent using vectors
Vector Applications in Euclidean Geometry
How to batch import Wavefront OBJ files? Is it possible to batch import Wavefront .obj to blender? I'm working on 2.69. If I select multiple .obj files, only one gets imported. Also if I hit A on file selection window to select all files (.obj and .mtl) I get an error, nothing gets imported. I have to import several hundreds of .obj into the blender file, is this possible? Hyphens in words with prefixes like neo-Nazism I have a question regarding the use of hyphens in words with certain prefixes. For example,     Neo-Nazism   Neo-Confucianism   Neo-liberal     are words with a hyphen, but     Neoplatonism   Neocolonialism   are words without hyphens. I'm especially confused with neo-Confucianism and neoplatonism: one contains  a hyphen and a capital letter and the other doesn't.  Is there a specific rule for hyphens in words with prefixes or is it purely arbitrary? Allow flags on locked posts How do I install python imaging library (PIL)? How do I install the python imaging library (PIL) on Ubuntu?  I tried doing  sudo apt-get install python-imaging   but now when i run     python selftest.py (a script I got from somewhere on the  website) I get (among other warning messages):  ... *** JPEG support not installed *** ZLIB (PNG/ZIP) support not installed ... *** 1 tests of 57 failed.   Have I somehow messed up the PIL -- how do I fix that?  Is maybe the PIL just fine, but that "selftest.py" is not really the right program for checking to see if PIL is installed properly -- how else can I tell if PIL is installed properly or not?  (I'm using Ubuntu 12.04 LTS "Precise Pangolin").  (What I'm ultimately trying to do is to add a 2D barcode generator to some python code, and all the 2D barcode generators I could find that were written in python all seemed to use the PIL). fsck error on boot: /dev/sda6: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY I dual boot Ubuntu and Windows 7. I was browsing on FireFox in Ubuntu and all of a sudden I get some error message, so I rebooted.  Now when I try to boot into Ubuntu I get this message:  fsck from util-linux 2.26.2 /dev/sda6 contains a file system with errors, check forced. /dev/sda6: Inodes that were part of a corrupted orphan linked list found.  /dev/sda6: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY.         (i.e., without -a or -p options) fsck exited with status code 4 The root filesystem on /dev/sda6 requires a manual fsck  Busybox v1.22.1 (Ubuntu 1:1.22.0-15ubuntu1) built in shell (ash) Enter 'help' for a list of built-in commands.  (initramfs) _   How can I fix this? I've tried to construct examples of rings that match all except one of the properties in the definition of a Dedekind domain. (This is an old number theory qual question from Berkeleys MGSA website).  The only starting point that I can think of would be  $R:=K[X_1,X_2,\ldots]$  for $K$ a field. This is clearly integrally closed as any polynomial relation is contained in $K(X_1,\ldots,X_n)$ for some large enough $n$ and $K[X_1,\ldots,X_n]$ is integrally closed being a UFD.  The problem is then to get rid of enough ideals from this ring $R$, so that its dimension would be 1, but I can't seem to figure out what to do.  Anyone have any other ideas for rings that could work as a starting point? The polynomial ring with an infinite number of variables is pretty much the only example that I know of for how to construct a non-noetherian ring that's still a domain. The swarms of dead men fighting for the good side always seemed wrong to me. I know they did it to be released from their plight, but.... It's one thing to curse somebody, it's quite another to back it up. How did Isildur back up his curse on the oath breaking men of Dunharrow? 1.  Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.  2.   A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)   3.   One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$). Align  elements side by side Word for feeling conflicting emotions simultaneously When is double-quoting necessary? The old advice used to be to double-quote any expression involving a $VARIABLE, at least if one wanted it to be interpreted by the shell as one single item, otherwise, any spaces in the content of $VARIABLE would throw off the shell.  I understand, however, that in more recent versions of shells, double-quoting is no longer always needed (at least for the purpose described above).  For instance, in bash:  % FOO='bar baz' % [ $FOO = 'bar baz' ] &amp;&amp; echo OK bash: [: too many arguments % [[ $FOO = 'bar baz' ]] &amp;&amp; echo OK OK % touch 'bar baz' % ls $FOO ls: cannot access bar: No such file or directory ls: cannot access baz: No such file or directory   In zsh, on the other hand, the same three commands succeed.  Therefore, based on this experiment, it seems that, in bash, one can omit the double quotes inside [[ ... ]], but not inside [ ... ] nor in command-line arguments, whereas, in zsh, the double quotes may be omitted in all these cases.  But inferring general rules from anecdotal examples like the above is a chancy proposition.  It would be nice to see a summary of when double-quoting is necessary.  I'm primarily interested in zsh, bash, and /bin/sh. How to apply CSS to iframe? I want to write several classes, most of them will use KOMA classes as base class.  However I stumbled upon the problem that some options (mainly concerned with layout) are not processed if given in the wrapper class:    \begin{filecontents}{test.cls}     \ProvidesClass{test}[2015/01/01]     \DeclareOption*{\PassOptionsToClass{\CurrentOption}{scrartcl}}     \ProcessOptions\relax     \LoadClass[paper=a5]{scrartcl}   \end{filecontents}  \documentclass{test}  \usepackage{blindtext}  \begin{document}   \section{Test}   \blindtext \end{document}   This results in   LaTeX Warning: Unused global option(s):     [paper=a5]  If i give the option as \documentclass[paper=a5]{test} it is used correctly, because I pass the options to scrartcl. However I want to change the defaults for the layout.   How can I achieve this? Is there a picture showing all the lengths used in lists? 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 Combinatorial Proof I have trouble coming up with combinatorial proofs. How would you justify this equality?  $$ n\binom {n-1}{k-1} = k \binom nk $$ where $n$ is a positive integer and $k$ is an integer. Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function? If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples? How to change the Google Chrome default print margins In Chrome, the default print margins are narrow. I would like the default to be wider.  Is there a way to make them wider? Why Windows sets new created file's "created time" property to old time? Unix epoch time to Java Date object I have a string containing the , and I need to convert it to a Java Date object.  String date = "1081157732"; DateFormat df = new SimpleDateFormat(""); // This line try {   Date expiry = df.parse(date);  } catch (ParseException ex) {   ex.getStackTrace(); }   The marked line is where I'm having trouble. I can't work out what the argument to SimpleDateFormat() should be, or even if I should be using SimpleDateFormat().