Differential equation problem. Integrating the logistic equation.
Modelling population with $\frac{dP}{dt}=P(\beta - \delta P)$
Strange character when using listings package Bluetooth from the command line in 12.04? I've been trying to pair up my bluetooth keyboard with my computer after reinstalling from a standard ubuntu 12.04 to a minimal install.  In the minimal install I have no gui, so I've been trying to use the various command line tools available, but I can't figure out how the pairing is supposed to go. Pairing when I had a gui worked flawlessly.  I've asked for help here:   No one has replied but a lot of details of my situation is available there.  How is one supposed to pair bluetooth devices from the command line? I'm using a PIN code to log on my computer. However, Num ⇩ is always turned off, even if on before shutting down my computer. This is quite annoying as I always have to retype my PIN.  I've tried the suggestions  (same as ). But with no result. If I don't enable num lock on log in screen and use the numbers above the normal keys, num lock turns on after log on. Then it's too late, unfortunately.  So, how can I make num lock enabled by default when logging in? Thanks.  Edit: While I still want to solve this, I've figured it is not such a big problem. I will need to press some key for the "enter PIN screen" to show, and using Num ⇩ works and enables num lock. But then, still annoying if the computer was only locked (with num lock on) and I turn it off, hehe. I'm currently studying  and Python programming. I just want to know how I would go about executing a Linux command in Python. The commands I want to execute are:  echo 1 &gt; /proc/sys/net/ipv4/ip_forward iptables -t nat -A PREROUTING -p tcp --destination-port 80 -j REDIRECT --to-port 8080   If I just use print in Python and run it in the terminal will it do the same as executing it as if you was typing it yourself and pressing Enter? Taylor expansion for $\arcsin^2{x}$ I stumbled upon this particular expansion that was included in  post.    $$ \displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$    This caught my eye because I remember trying to derive a Taylor series for $\arcsin^{2}(x)$ a while ago without much success.  Can anyone prove this or point me to a material that would show a proof of this identity?  EDIT :  Feel free to use any mathematical apparatus at hand. I'm not interested in a proof fit for a certain level, nor am I looking for utmost elegance (though that would be lovely). Prove that for for all $p,q\in \mathbb{Z}$, $q&gt;0$ we have: $$ \left| \sqrt{2}-\frac{p}{q} \right| &gt; \frac{1}{3q^2}. $$  To be honest, I do not know where to start - any help would be appreciated. Let $\alpha$ be an irrational real number and let $a_j$ be a sequence of rational numbers converging to $\alpha$. Suppose that each $a_j$ is a fraction expressed in lowest terms: $a_j = \alpha_j / \beta_j$. Prove that the $\beta_j$ tend to $\infty$  Attempt: AFSOC that $\beta_j$ does not tend to infinity, then it is bounded by some $M$. We can also find an interval $\alpha \in (k, k+1)$. Let $\epsilon &gt; 0$, $\exists N_0$ such that for all $N &gt; N_0$ we have $|a_N - \alpha|  &lt; \epsilon$. So $|\frac{p_N}{q_N} - \alpha| &lt; \epsilon$. We try to use the fact that $q$ is bounded but fail to derive a contradiction. I do not know how to use the fact that $\alpha$ is between two integers, although it might not be relevant at all.  Ideas? "Where" exactly are complex numbers used "in the real world"? I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be used/applied and hence am curious.  So what practical applications of complex numbers exist and what are the ways in which complex  transformation helps address the problem that wasn't immediately addressable?  Way back in undergrad when I asked my professor this he mentioned that "the folks in mechanical and aerospace engineering use it a lot" but for what? (Don't other domains use it too?). I'm well aware of its use in Fourier analysis but that's the farthest I got to a 'real world application'. I'm sure that's not it.  PS: I'm not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. A naive analogy is deciding the height of tower based on trigonometry. That's going from paper to the real world. Similarly, what is it that is analyzed in the complex world and the result is used in the real world without imaginaries clouding the problem?  The question:  is nice but covers a more mathematical perspective on interim results that make solving a problem easier. It covers different ground IMHO. How do I compare two string variables in an 'if' statement in Bash? I'm using X11 forwarding over ssh to run Linux apps on my Windows box, and when the network drops it loses everything that was running. Is there anything similar to screen for X11? C++ Redefinition Header Files (winsock2.h) Why won't QGIS recognize my GPS? How to insert an item into an array at a specific index (JavaScript)? Why does a relay have a minimum applicable load? I am trying to find relays for my application and I read  which looks fine but specifies     Minimum applicable load: 10mV 10µA   In my circuit, it is expected that the relay closes but no voltage and current is applied. You can think of it as 2 relays in series where one is open and one is closed, so there is no current. This sounds like something I'd do since I was at school.  Why would a relay require a minimum voltage and current on the load side? Is it allowed to operate that relay under my conditions or not? What could possibly break in a relay if I don't respect this requirement? What does "minimum applicable load" mean? When and how do I need to consider this value? Degrees of freedom for standard deviation of sample would someone please explain why the degrees of freedom for a random sample is n-1 instead of n ?   I'm looking for an explanation that is intuitive and easily understood by a high school student. How are Dementors created/born? This is inspired by some discussions in .  In the Harry Potter universe, how are Dementors born?  Or are they artificially created by some magical means? how to get the base url from jsp request object? Why is the "Arnold Schwarzenegger" terminator referred to by two different product numbers (T-101, T-800)? Let $\alpha(t)$ be a regular curve. Suppose there is a point $a$ in $R^3$ such that $\alpha(t)-a$ is orthogonal to $T(t)$ for all t. Prove that $\alpha(t)$ lies on a sphere.  So, I let $\alpha(t)=(x(t), y(t),z(t))$ and $a=(a,b,c)$.  Since, $\alpha(t)-a$ is orthogonal to $T(t)$,  $&lt;\alpha(t)-a,T(t)&gt;=0$.  But I don't know how to solve this.. Is there offside when the ball deflects off an opponent? I was refereeing a 10 year old's junior game today and this situation happened. The home team's defender attempted to  pass from their own half to their forward who was in an offside position beyond the halfway line.  I was going to blow the whistle but the ball deflected off an away defender and then went to the originally offside attacker who put the ball wide.  I viewed this as onside at the time because the ball struck the away team player, however now I think it was offside because at the time of pass, the home player was offside.  Is this offside or not?