I'd like to solve functional equation: $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}.$  I've managed to get: $f(0)=0,f(n)=0$ for all $n\in N$; $f(\frac{1}{2})=0$; $f(-x)=-f(x)$. I'll be grateful for any help.
I have the following functional equation. Find all continuous functions $f:(-1,1) \to \mathbb R$ such that $$ f(x+y)=\frac{f(x) + f(y)}{1 - f(x)f(y)} $$ The first obvious solution is $f(x) \equiv 0$. Another one I guessed, it is $f(x) = \pm \tan x$. I suspect, that there are no more solutions. The problem is that I don't know how to prove that.  Since we have a rational equation, I have no idea how to make any substitutions in order to get expression for $\tan x$ (as it is not a rational expression).  P.S. I can also show that it is true that (a) $f(0) = 0$ (take $x=y=0$), and (b) $f(-x) = -f(x)$ (take $y=-x$).  Update: there is a solution of this problem also here:
VirtualBox 5.0.40 crashes the whole host since update to kernel 4.13.0-26-generic Compute the integral $\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$ "Congratulate for" vs. "congratulate on" Which is correct?     I congratulated him for coming first in the race.      I congratulated him on coming first in the race. Check if macro is fully expandable Question  Is there a way to check whether a macro is fully expandable (or rather "safe in an expansion-only context" )?    Consider this code:  \def\a{Just a string} \def\b{\a} \def\c{\def\unsafe} \def\d{\c}   How could I check which of the macros (a-d) are safe in an expansion-only context? By looking at them I know that a and b are whereas c and d are not but if I wanted to know the same for a macro I haven't written myself this could get quite useful.      Background  I am working on a way to detect whether some input is a valid number in PGF. For this I developed  which makes use of passing the input into \pgfmathfloatparsenumber. The problem I have run into is that said macro appears to somehow manages to expand the input until there is an error (if the input is in fact not safe in an expansion-only-context). I tried using protected, noexpand and similar but somehow PGF manages to circumvent those.    So the idea is to check whether the input is safe before actually passing it to PGF. The problem is: I don't know how I'd go about that... Undefined reference error for template method I would like to clarify something that I don't think is clear in   The first automatic delete rule kicks in at 9 days for closed questions:     If the question was closed more than 9 days ago, and ...         was not closed as a duplicate   has a score of 0 or less   has no answers with a score > 0   some other criteria that don't matter right now...      Then, at 30 days, there's:     If the question is more than 30 days old, and ...         has −1 or lower score   has no answers   is not locked      This rule applies to both open and closed questions, but it doesn't mention the same exemption for duplicates.  I'm pretty sure that I've seen old, downvoted duplicates disappear on ServerFault (and that fits what the rule says), but I'm also sure I've seen lots of comments along the lines of  "duplicates aren't deleted."  So what I'm after is a clear statement from a power-that-be or a clear reference that says "duplicate questions fall under the same 30-day automatic deletion rule as any other question."  (or that they don't)  (Note: I realize this doesn't apply the stub left behind when a question is merged into another.  Those "stubs" are locked and are never deleted.) "It is not possible to check out a single file. The finest level of checkouts you can do is at the directory level."  How do I get around this issue when using Subversion?  We have this folder in Subversion where we keep all our images. I just want to check out one file (image) from that. This folder is really big and has ton of other stuff which I don't need now. Can I clear an area of my prison for everyone BUT max security prisoners? My Bridgestone MB-5 currently has a 7-speed rear wheel. I'd like to convert it to a 10-speed. Is this possible? Is it practical? I have noticed this on many different Windows machine, on many different occasions: the RAM usage reported by Task Manager or Resource Monitor often seems to add up to an amount which is significantly lower than the actual amount in use.  For example, many times on my laptop or my desktop, I have seen something like 7GB in use, and yet the total of the Working RAM Set is more like 3GB. I just can't figure out where it's being used!  Here's an extreme example I noticed today in Resource Monitor on a server:    If you right-click the image and open in a new tab, and view the numbers, you'll notice that the Working Set (which doesn't include nonphysical virtual memory) add up to about 1.7GB. I get similar numbers by adding up RAM usage in Task Manager when "Show processes from all users" is enabled.  Now here's a screenshot of task manager's Performance tab:    This says 7.6GB of physical memory is in use.  I see this all the time, on personal computers, laptops, and now servers: the total RAM usage reported by system tools only accounts for about 1/4 of the RAM usage I observe. WTF is going on???  Is there any satisfying explanation of where all my RAM is? What is gobbling it up, and why is it leaving no trace?  EDIT: Here's a picture of the graphical RAM use, as user whs asked for:    EDIT 2: In response to James' response, here's a picture of the nonpaged processes in poolmon.exe, sorted by size:    These results confuse me. poolmon correctly states that I have 6GB of nonpaged pool in use, but all the nonpaged pool processes are less than 8MB in size.  What could this mean? Is poolmon failing to detect some of the processes using the nonpaged pool? A simple pendulum moving at a relativistic speed - how does the period change? I've been pondering the precise mechanism of time dilation for the example of a simple pendulum in two different situations:   The observer and ground are at rest in one frame of reference;  the pendulum is moving at high speed with respect to that frame. The observer is at rest in one frame of reference; the pendulum and the ground together move at high speed with respect to that frame.   user8260 has pointed out that in situation 1, in the pendulum's frame $g$ is greater by $\gamma^2$ compared to $g$ in the observer's frame. Thus in the pendulum's frame the period is less than it is in the observer's frame by a factor of $1/\gamma$, just as one would expect from time dilation.  But what about situation 2? Here, compared to the pendulum frame, the observer sees the pendulum with the same length in a stronger gravitational field, yet observes a longer period. Does the inertial mass of the pendulum change differently than its gravitational mass? Also, does the analysis depend on whether the plane of swing is perpendicular or parallel the direction of motion? I was trying to install Lightworks 11.5.1 (amd64) on Ubuntu 14.04(amd64) but there's a notice: Dependency is not satisfiable: libtiff4. Is there anyway to solve this dependency issue?  Thanks in advance. What happens to the Windows 7 key when upgrading to Windows 8? Microsoft is offering "Microsoft Windows 8 Professional Upgrade" at a discounted price compared to the OEM version. However it states:     If you currently have a personal computer running Windows 7, Windows XP or Windows Vista then you can upgrade to Windows 8 Pro (Professional).   My question is what happens to the Windows 7 key that I use to upgrade? Will it become the Windows 8 key or will it generate a new Windows 8 key?  Basically if I upgrade using my Windows 7 key from my laptop for my desktop will I be able to continue using the laptop on Windows 7 while concurrently using Windows 8 upgrade on my Desktop? Fastest way to clamp a real (fixed/floating point) value? Is there a more efficient way to clamp real numbers than using if statements or ternary operators? I want to do this both for doubles and for a 32-bit fixpoint implementation (16.16). I'm not asking for code that can handle both cases; they will be handled in separate functions.  Obviously, I can do something like:  double clampedA; double a = calculate(); clampedA = a &gt; MY_MAX ? MY_MAX : a; clampedA = a &lt; MY_MIN ? MY_MIN : a;   or  double a = calculate(); double clampedA = a; if(clampedA &gt; MY_MAX)     clampedA = MY_MAX; else if(clampedA &lt; MY_MIN)     clampedA = MY_MIN;   The fixpoint version would use functions/macros for comparisons.  This is done in a performance-critical part of the code, so I'm looking for an as efficient way to do it as possible (which I suspect would involve bit-manipulation)   EDIT: It has to be standard/portable C, platform-specific functionality is not of any interest here. Also, MY_MIN and MY_MAX are the same type as the value I want clamped (doubles in the examples above). What is the next perfect square of the form 14444... in decimal notation? We know that $12^2 = 144$ and that $38^2 = 1444$. Are there any other perfect squares in the form of $\frac{13}{9} (10^n - 1) + 1$ (i.e. $1$ followed by $n$ $4$'s), and how would we prove it? Adjectives and nouns to describe an answer that doesn't answer the question? I've heard of the adjectives 'nonresponsive' or , but are they restricted to law^?     Collins Dictionary:  =       not responsive, not reacting to a stimulus   The definition above differs from the definitions of adjectives like 'equivocate,  evade' and nouns like , but these are negative and accusive; I prefer neutral connotation.  ^Only if you wish, please see: ,  by David Berg,  by Thomas A. Mauet. What is the correct vector input for a texture connected to a Cycles lamp so that I can make a projector-like textured light?  I am trying to make an projector-like effect with a textured light in Cycles, currently trying it out with a Spot type of lamp, but I am open to any solution involving other lamp types if they work correctly.  The problem is that whatever vector input I use, the texture is always projected downwards, towards the global $-Z$ axis in scene world space, no matter where I point my spot light. I would like to have the texture always project "forward" towards the the direction the spot light is pointing, in the local $-Z$ axis.    The plan would then be to use this spot at several places in my scene, pointing at different directions, and as it currently stands I would have to have a different node setup for each spot instance, manually compensating for each unique rotation in the scene, which would quickly become unpractical.  I have searched around the internet and I am currently using a solution similar to the one on this .  Is there any better method, perhaps involving some funky Vector Math, or is this not currently supported? Change the display scaling on the fly I'm looking for a keyboard shortcut or a CLI instruction to change the display scale on the fly. Seems there's nothing in compiz-config manager to handle this? Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction? I oversaw a high-school mathematics test the other day, and one of the problems was the following     Show, using induction or other means, that $$\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$$   The induction proof is very standard, where the induction step relies on the fact that $\frac{1}{n+1} + \frac{1}{n(n+1)} = \frac{1}{n}$, and I'm sure it's been answered on this site before. However, I got intrigued by the "or other means" part of the question.  I don't know whether the teacher who wrote the test even considered any alternative solutions (he may just have written it so that if anyone has a crazy idea that works out, then they should get a full score for it), but I tried to find one. For instance, we may do the following telescope-ish argument: $$ \frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \cdots + \frac{1}{(n-1)n} + \frac{1}{n(n+1)} + \frac{1}{n+1}\\ = \frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \cdots + \frac{1}{(n-1)n} + \frac1n\\ \vdots\\ = \frac{1}{1(1+1)} + \frac12\\ = 1 $$ However, I feel that this is just an induction proof in disguise (hidden in the vertical dots). If one uses the mechanics of the induction proof to check whether the formula is true for a specific $n$, then one certainly does the exact same calculations as I have done here.  Is there a proof of this fact that clearly does not use induction (or at least hides it better)? The more elementary the better, and the ultimate goal would be to do it within the syllabus of the students taking the test (or at least not far from it). For reference, they should be familiar with the summation formula of arithmetic and geometric series and their derivations (so techniques resembling those would be well within specifications).  If there is a solution using calculus, then the students should be able to integrate elementary trigonometric functions, as well as exponential functions, logarithms and rational functions. They are familiar with integration by parts, substitution and partial fractions.  I welcome more advanced solutions as well, of course. Is there a graceful way to handle passing a list of ids as a parameter to a stored procedure?   For instance, I want departments 1, 2, 5, 7, 20 returned by my stored procedure. In the past, I have passed in a comma delimited list of ids, like the below code, but feel really dirty doing it.   SQL Server 2005 is my only applicable limitation I think.  create procedure getDepartments   @DepartmentIds varchar(max) as   declare @Sql varchar(max)        select @Sql = 'select [Name] from Department where DepartmentId in (' + @DepartmentIds + ')'   exec(@Sql)