Why is a satellite's GPE negative?
Why is potential energy negative when orbiting in a gravitational field?
Is a current transformer appropriate for a voltage step up flyback converter? I am designing a flyback converter, input 170V output 1200V at 2mA.  I am having trouble finding an appropriate transformer, most do not have a high enough N. Current transformers seem to have ranges in 1:500, so I could work with that.  Is it proper to use a current transformer as a conventional step up voltage transformer? How to get glare on "Quick Smoke" (Fire & Smoke explosion) A word that says a person is both female and your friend Why doesn't bash expand this variable when I prefix a command with a "one time variable assignment" If I run this bash command and prefix the statement, so that the variable fruit should exist, but only for the duration of this command:  $ fruit=apple echo $fruit  $   The result is an empty line. why?  To quote a comment from wildcard on :     parameter expansion is done by the shell, and the "fruit" variable is   not a shell variable; it's only an environment variable within the   environment of the "echo" command   An environment variable is still a variable, so surely this should still be available to the echo command? Should a retrieval method return 'null' or throw an exception when it can't produce the return value? Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Is the converse true? I.e., if $g \circ f$ is invertible, does it follow that $f$ and $g$ are both invertible?  I know that if $f$ and $g$ are invertible, then they are both injective and surjective. Therefore, I assume from this that I have to show $g \circ f$ is injective and surjective as well, thus proving it is invertible. However, I don't know how to show $g \circ f$ is bijective based only on the information I have. How to create quizzes with macqtex? Using X11 in OSX 10.8.3 : "ld: library not found for -lX11" I am just curious that the following format looks good to you or not? The preamble I used is just   \documentclass[12pt,a4paper]{article}  \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb}   I don't feel it's so good because the first paragraph has no indent but the second does. If the first paragraph is much longer, than it's fine. But if not, it looks strange. Should I leave it, or you have suggestions for this? $ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $ $$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $$   (can this be duplicate? I think not) I tried it using many methods  $1.$ Solve this conventionally taking $1^\infty$ form in no luck  $2.$ Did this, expand $ {(1+x)^{\frac1x}}$ using binomial theorem got $\frac13$ then grouped coefficients of $x^0$ and it cancelled with $e$ then took coefficient of $x$ cancelled with $\frac{ex}{2}$ and so on   very messy right ? at last I got $\frac13$ but that's not the expected answer!  I must have went wrong somewhere can anyone help me with this. USB is a popular standard and I was wondering whether there is some way to connect two or more computers using USB ports rather than the Ethernet ports. Find $n$ such that $n/2$ is a square, $n/3$ is a cube, and $n/5$ a fifth power My equation is not showing big bracket and not showing in next line Given fields $M/E/F$, why does $[M:F] = [M:E][E:F]$? Let $M$ be a finite extension of $E$ and let $E$ be a finite extension of $F$. Then $M$ is a finite extension of $F$ and $[M:F] = [M:E][E:F]$.   Is there an easy explanation and/or proof for this theorem? My instructor gave an incomplete proof in class but got stuck and I'm kind of confused of what's going on. How to find distance from the latitude and longitude of two locations? opengl storing multiple indices into indices buffer After parsing collada files I found out that we have to load two(or more) indices each pointing to say a vertex or normal etc. like this (&lt;p>3 0 2 0 0 0&lt;/p>)  Is there a way to load (and use)these indices to indices buffer(ELEMENT_ARRAY_BUFFER) like this (vnvnvn...) ; where v is vertex pointer, n is normal pointer from the above. with vertices loaded to VBO1 like (vvvv...) and in VBO2 like (nnnn...). So that we dont have to repeat the normals in VBO2.  VBO like (vnvnvn...) is also ok. (and not something like this ), I'm not trying to use multiple indices buffer. I'm trying to put all the given indices into one index buffer and use them alternately.  (If I wasnt clear i will try for more clarification.) Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$. Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance! How many solutions for an equation with simple restrictions I'm working on an assignment in which I have to count the number of solutions for this particular equation: $$x_1+x_2+x_3+x_4=20$$for non negative integers with $x_1&lt;8 $ and $x_2&lt;6$   I'm aware that this kind of a task isn't that complicated, but I don't get combinatorics in general that well.  So I've tried two following approaches to get this done.  Firstly I tried to substitute the variable x:  $x_1+x_2+x_3+x_4=20 \Leftrightarrow y_1+y_2+y_3+y_4=34$  in which $y_1=x_1+8$ and $y_2=x_2+6$ (casue $x_1=y_1-8$ and $x_2=y_1-6$) Following this approach the total number of possible solutions would be  $${34+3 \choose 3} $$  But I'm not sure if its the right solution.  The second approach is to sum all of the possible values that $x_1$ and $x_2$ could possibly take, also $x_1=0,1,2,3,4,5,6,7$ and $x_2=0,1,2,3,4,5,6$ And then  count all the possibilities for each of the variables $${20 -x_1-x_2+1\choose 1}$$ and sum them together like this: $${21\choose 1}+{20\choose 1}+{19\choose 1}+{18\choose 1}+... $$ and so on...  I'm sure I'll get the correct number with this one, but I'm not feeling like summing all of this possibilities. There's got to be a better, more elegant way to deal with this.  My professor gave me a hint that I should do it using the complement. Can the relative pronoun "whose" be used for animals, things and countries? When I was a student of English as a foreign language, more years ago than I care to count, I was taught that the relative pronoun “whose” could only be used for human beings, i.e., when someone possesses something.   My teachers were native speakers from England, then.   Since that time, I’ve seen “whose”, as a relative pronoun, used in several contexts where no human beings are mentioned, e.g. with animals, objects, countries, abstract nouns, etc, in American newspapers and magazines. Unfortunately, I can’t remember any of the sentences and cannot give examples.  What is acceptable in written and spoken English, and what isn’t, concerning the use of “whose” ? "A friend of my father" vs. "a friend of my father's" — which one is right? Why? I have seen them both in my reference book, which is written by a native speaker. I wonder why there are two different phrases to describe the same meaning and am looking forward to some reasonable details to explain why this is so.