$f (f(x))=-x$ - there's no such function
Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
Contracting with Levi-Civita (totally antisymmetric) tensor What factors influence the spring constant? For certain springs we have been taught that Hookes law, $F=-kx$ works. For such springs, what factors determine k. For instance, k is proportional to length. Can k be expressed in terms of dimensions, mass and other fundemental quantities, or is k a fundemental property of the spring itself? 80s/90s book where boy stumbles on game where aliens gamble their planets I don't remember much about it other than the setting. It's about a boy who one night stumbles upon some different alien races playing a game in which their civilizations will end if they lose. He gets involved in the game somehow and is hunted. Equivalent Resistance? How do you find the equivalent resistance here? (I realized it's a Wheatstone bridge, but it's unbalanced, so that doesn't help.) I tried using Kirchhoff's laws, but there is no cell given. findstr /v "black"  File1.txt   Above DOS command will display content of 'File1.txt' which are not matching string "black".  How to modify this command , if I need to filter words "black" and "white" ? Why is voltage added when batteries are connected in series but not in parallel? As I understand, the different voltage between two object is caused by the total different charged between two object. Thus, the higher extra electron between two opposite charged object, the higher voltage between them, isn't it? So if i have two battery or capacitor connect in parallel, the negative connect to negative, the positive connect to the positive. Caused the charge from each plate to be double. So that's as my expected, it must be double voltage, but ironically why it's out of my expected and the reality is vice-versa? Non-Affine image transformations in .NET Are there any classes, methods in the .NET library, or any algorithms in general, to perform non- transformations? (i.e. transformations that involve more than just rotation, scale, translation and shear)  e.g.:      (source: )      Is there another term for s? Why does change in speed of a wave make it refract? When a light wave enters a medium with a higher refractive index (e.g. from air to standard glass) and its speed decreases, why does that make it refract/bend?  I understand that wavelength decreases and frequency stays the same and therefore its speed decreases, but I can't find anywhere whatsoever why the speed decrease cause the wave to refract. So could someone please explain this? How to install the latest versions of NodeJS and NPM? I noticed over at the  website that node is currently at v 0.12.0.  Can someone let me know how to install the latest version of node together with npm (terminal commands please)? Proving a basis for inner product space V when $||e_j-v_j||< \frac{1}{\sqrt{n}}$. Suppose $(e_1,e_2,...,e_n)$ is an orthonormal basis of the inner product space $V$ and $v_1,v_2,...,v_n$ are vectors of $V$ such that $$||e_j-v_j||&lt; \frac{1}{\sqrt{n}}$$ for each $j \in \left\{1,2,...,n \right \}$. Prove that $(v_1,v_2,...,v_n)$ is a basis of $V$.  I am completely lost and just starting to learn about inner product spaces. Could someone provide a proof with the explanation of how you got there? Suppose f is a function such that $f'(x) = \frac{1}{x}$ for all $x &gt; 0$. Prove that if $f(1) = 0, then f(xy) = f(x) + f(y)$ for all $x,y &gt; 0$  As stated above, how do i go about doing this question?   Hi all, i would like to apologize if the question is missing context or other details.   This question was part of my calculus tutorial. I was on the topic of application of differentiation when this question was given to me.  Thus far i have learnt about intermediate value theorem, mean value theorem, rolle's theroem, Fermat's thereom.  As of this point in my learning, i have not touched upon integration. The only reason why i accepted the answer with integration in it was because i understood it.  Thus i believe i am suppose to use one the few theorems that i've stated to solve this question. But i have no idea where to start.  At first i differentiated the equation $f(xy)=f(x)+f(y)$ with respect to x  i believed the answer to be $f'(xy)\frac{dy}{dx}=f'(x)+f'(y)\frac{dy}{dx}$  which would be $\frac{1}{xy}\frac{1}{x}=\frac{1}{x}+\frac{1}{y}\frac{1}{x}$  At this point a few questions came to my mind  1) am i suppose to differentiate the eqn in the first place 2) what has $f(1)=0$ got to do with anything 3) what theorem am i suppose to use to solve this question  This was my thought process throughout the question and i hope someone would point out my mistakes. Thank you Can "cattle" be singular? I've grown up on a farm, and my dad and his dad, apparently, always used "cattle" to refer to both the singular and plural forms of the domestic bovine. I've always assumed it's how the word "deer" is.  However, I've heard people say that this is incorrect and the singular is just "cow", but this has always offended us as a cow is a mother cattle, and is incorrect if you're referring to a steer, a bull, or a heifer.  So, is cattle singular as well as plural? If not, is there some general, non-gender-specific word that should be used instead? I've tried searching for this question but couldn't find it on stackexchange. This is a common type of interview question; I ran into it doing brain teasers on a probability puzzles app, and if you fine people agree with my logic, I will inform the app developer that his/her answers are incorrect.   The problem is essentially this:  You are sentenced to death for thievery. The King is magnanimous and decides to put your fate in the hands of chance. You are given $100$ white marbles and 100 black marbles, and $2$ urns. The king will choose an urn at random and pull out a single marble at random; if the marble is white, you live, if its black, you die. If you place the marbles in the best way possible, what is your probability of survival?  I started with the base case: $100$ white marbles in one urn, $100$ black marbles in the other. This comes down to a $50$-$50$ chance of survival. I then worked my way to deciding that placing $1$ white marble in one urn and $99$ white marbles + $100$ black marbles in the other urn would be the "best way possible", which yields the following:  $$P(\text{Survival}) = \frac{1}{2}(1+\frac{99}{199}) \approx .749$$  Selecting $1$ of $2$ urns at random gives $\frac{1}{2}$, the urn containing $1$ marble gives $1$, and the other that contains $99$ white marbles and $100$ black marbles gives $\frac{99}{199}$ because there are $99$ possible white marbles to select out of $199$ total marbles.  The app claims that the correct answer is $\frac{1}{2}(1+\frac{99}{200}) \approx .748$   I see where the $200$ comes from, but I do not think it is right to say that there are $200$ marbles in the other urn. Who is correct? How do I go about porting Ubuntu Touch to different devices? What does "sudo apt-get update" do? Why comment reply auto-completion feature (@) sometimes doesn't work? Fibonacci addition law $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$   My (very limited) attempt so far: after creating a small list of the values $F_0=0, F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, F_6=8, F_7=13, F_8=21, F_9=34, F_{10}=55$ i can see that yes it does seem to work for instance $F_{6+3}=F_5 F_3 +F_6 F_4 = 10 +24 = 34 = F_9$. However, I really don't know where to begin as showing that this must hold in general terms. Should I be looking to use limits? Or perhaps induction? What is the best way to solve this? Running programs in the background from terminal Mortarboard badges being awarded as a result of bounties after recalc How to auto populate with conditions?