Why cp and mv behave differently in bash script than command prompt?
Why doesn't my Bash script recognize aliases?
What are the biggest changes between UA Ranger and PHB Ranger? A geometric reason why the square of the focal length of a hyperbola is equal to the sum of the squares of the axes? When I teach conics, I give a simple geometric argument for the Pythagorean relation for ellipses. (By "Pythagorean relation" I mean the fact that the square of the focal length $c$ is the difference of the squares of the semimajor and semiminor axes, i.e. $c^2=|a^2-b^2|$ for the ellipse defined by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.) The picture argument goes as follows: just draw the obvious isosceles triangle from the two foci to one of the two points at the end of the minor axis. We see two right triangles; in each, one leg is the focal length $c$ and the other leg is the semiminor axis, while the hypotenuse is the semimajor axis. The result follows from the Pythagorean theorem applied to this right triangle.  But when it comes to the Pythagorean relation for hyperbolas, I can't think of an equally convincing picture to draw. Instead I have to give a rather inelegant algebraic argument for the relation $c^2=a^2+b^2$ (for the hyperbola defined by $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, with focal length $c$), directly from the definition of the hyperbola. The messy details are below.*  Now, obviously there is an asymmetry between the argument for the ellipse and the argument for the hyperbola: the ellipse argument is almost trivial and doesn't involve working straight from the locus definition, but the hyperbola argument is tedious and does require working straight from the definition. So, my question:     Is there an equally persuasive picture I could draw to convince high   school students of the Pythagorean relation for hyperbolas -- without having to get technical with points at infinity, say?   The natural triangle to draw is the right triangle with legs from the center to a vertex and from that vertex to an asymptote, which has legs of length $a$ and $b$; but then the problem becomes why the length along the asymptote should be equal to $c$. I doubt there is a purely (Euclidean) geometrical reason because this triangle lies on the asymptote, not on the hyperbola itself; I suspect some sort of limiting procedure will be required.    [*] The tedious algebraic details:     The foci lie at $(\pm c,0)$.   Say $a&gt;b$. Then hyperbola is the locus of points $(x,y)$ such that       $$\left|\sqrt{(x+c)^2+y^2}-\sqrt{(x-c)^2+y^2}\right|=2a$$      Assume for simplicity the expression between the absolute value bars   is positive. (The algebra is even more tedious if we don't make such a   simplifying assumption.) Then moving the right root to the other side   and squaring gives      $$(x+c)^2+y^2=4a^2+(x-c)^2+y^2+4a\sqrt{(x-c)^2+y^2}$$      Simplifying gives      $$xc-a^2=a\sqrt{(x-c)^2+y^2}$$      and upon squaring again we have      $$x^2c^2-2xca^2+a^4=a^2(x^2-2xc+c^2+y^2)=a^2x^2-2xca^2+c^2a^2+a^2y^2$$      which reduces to      $$x^2(c^2-a^2)-y^2a^2=a^2(c^2-a^2)$$      Defining $b^2:=c^2-a^2$ brings the equation into the form   $x^2/a^2+y^2/b^2=1$, and the Pythagorean relation follows from the   definition of $b$. Subgroups of Symmetric groups isomorphic to dihedral group Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n&gt;2$? Activity Monitor in Mavericks "Show Memory Usage" dock icon gone? I just purchased a new MacBook Pro with Mavericks installed and noticed that Activity Monitor doesn't have the "Show Memory Usage" dock icon option anymore.  I really liked this option since I used it to prevent disk swapping by closing applications when I had too many open.  What gives? I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$  have real part smaller than 1, but I don't know how to prove it. If Re(z) > 1 than  ${\zeta}( z ) = \sum_{n = 1}^{\infty} \left( \frac{1}{n} \right)^z$ How do I iterate over a range of numbers defined by variables in Bash? How do I iterate over a range of numbers in Bash when the range is given by a variable?  I know I can do this (called "sequence expression" in the Bash ):   for i in {1..5}; do echo $i; done   Which gives:     1    2    3    4    5   Yet, how can I replace either of the range endpoints with a variable? This doesn't work:  END=5 for i in {1..$END}; do echo $i; done   Which prints:     {1..5} How do I programmatically log in a user with a POST request? Can you recommend a Java library for reading (and possibly writing) CSV files? On all of the other SOFU sites a downvote means The question is unclear or not useful.  However, here on meta, a downvote has become a sign of I don't agree with the question, suggestion, feature request, bug, etc. being described in the question.   What does a downvote on Meta really mean? What other reasons do people downvote on Meta? Why is it okay for meta to redefine the meaning of the downvote?  The tooltips and instructions are all the same here on meta.   Note: I'm not saying I disagree with the practice, just curious. There is an option to filter only one color (R,G,B,Y) in my camera. What I want to know is can this effect be achieved in photoshop? And what is this called? I tried googling monochrome and came up with nothing. My camera calls it part color. How to add PlotLegends in a Show function? Replacements for switch statement in Python? Unable to make recursive call in Javascript Do we really need reals? It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions.  Since almost all real numbers are not computable, it seems that real numbers are a set too big and most of them are unnecessary.  So why do mathematicians love this plethoric set of numbers so much? Is this defined by the language? Is there a defined maximum? Is it different in different browsers? How does bash interpret the equal operator with no surrounding spaces in a conditional? The following script does not behave as I would have expected.  Adding spaces around the '=' in the conditional made it perform how I wanted, but it got me thinking, what is it actually doing inside the conditional?  #!/bin/bash S1='foo' S2='bar' if [ $S1=$S2 ]; then     echo "S1('$S1') is equal to S2('$S2') fi echo $S1 echo $S2   The output is:  S1('foo') is equal to S2('bar') foo bar   The contents of S1 and S2 don't change from what they are assigned, so the = doesn't perform an assignment. Massive spam users and spam entries despite triple layer protection Winter Bash 2014 closing remarks - ideas for next year we had a question like this, and  SE  members have supported one this year, and nobody seems to have made one yet.  Hence:  What does everybody want for next year's Winter Bash? It's okay to repost old ideas that have already come up if you feel they haven't had enough attention (but please don't go just copy/pasting every  !).  A few ideas I can think of off the top of my head:   Moar hats! There's been plenty of time to get all the hats this year; it would be great if next year there'd be more to do. Keep secret hats secret-er. This could be pretty hard to do, but possibly by deleting/locking "how do I get secret hats" posts?   One idea per answer, please! Is there a closed formula for this summation? I have the summation  $$\sum\limits_{i=1}^n \frac{1}{i^2}$$  And I don't know how to find a closed formula for it. Any ideas? Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.