I’m trying to see what instruments I could use to analyse the following ODE in the complex plane: $ \dot{z} = \exp(it)\cdot \bar{z}$. Where $z$ is a function of real valued time.  ${{}}$
How do I solve $$x'=e^{it}\overline{x}?$$  This is a complex differential equation, but I don't see how to solve it.  Edit: the original ODE is given by $$(x', y')=\begin{pmatrix}\cos t&amp; \sin t\\ \sin t&amp;-\cos t\end{pmatrix}(x,y)$$ I want to show that solutions of this ODE do not remain bounded for all $t$, and the idea was to solve the complex equation.
SSH remote user need to be logged in? Enable Wireless Monitor mode for traffic sniffing I need to sniff all the wifi traffic that I can capture using wireshark in Ubuntu. My wireless interface is named as eth1 in the interfaces list. When I select that interface in Wireshark and start new sniff, I got following error;    As  thread says, I tried to installed the mentioned driver but I got an error in installation saying that that driver is not supported by my device. After that I tried the second answer in the same thread and run following command to enable monitor mode in my wireless card.  echo 1 &gt; /proc/brcm_monitor0   Then I saw a new Ethernet interface (not a wireless interface ) called prism0 in wireshark interface list. I couldn't start a sniff using that interface using monitor mode because in that interface settings, monitor mode check box has been disabled.  I have installed a broadcom wireless driver in Additional Drivers as follows:    Is there an error listing my Wifi interface as eth1 rather than wlan0 in the interfaces list? How can I start my wireless interface in monitor mode for sniffing?   Thank you. Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation   $$\theta:U^* \otimes V^* \to (U \otimes V)^*$$  defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$  I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck. Back transforming regression results when modeling log(y) I'm fitting a regression on the $\log(y)$.  Is it valid to back transform point estimates (and confidence/prediction intervals) by exponentiation?  I don't believe so, since $E[f(X)] \ne f(E[X])$ but wanted other's opinions.  My example below shows conflicts with back transforming (.239 vs .219).  set.seed(123)  a=-5 b=2  x=runif(100,0,1) y=exp(a*x+b+rnorm(100,0,.2)) # plot(x,y)  ### NLS Fit f &lt;- function(x,a,b) {exp(a*x+b)}  fit &lt;- nls(y ~ exp(a*x+b),  start = c(a=-10, b=15))  co=coef(fit) # curve(f(x=x, a=co[1], b=co[2]), add = TRUE,col=2,lwd=1.2)  predict(fit,newdata=data.frame(x=.7)) [1] 0.2393773  ### LM Fit # plot(x,log(y)) # abline(lm(log(y)~x),col=2) fit=lm(log(y)~x) temp=predict(fit,newdata=data.frame(x=.7),interval='prediction') exp(temp)         fit       lwr       upr 1 0.2199471 0.1492762 0.3240752 Why does (0 < 5 < 3) return true? Prime dividing the binomial coefficients It is quite easy to show that for every prime $p$ and $0&lt;i&lt;p$ we have that $p$ divides the binomial coefficient $\large p\choose i$; one simply notes that in $\large \frac{p!}{i!(p-i)!}$ the numerator is divisible by $p$ whereas the denominator is not (since it is a product of numbers smaller than $p$ and $p$ is prime).  My problem is with generalizing this argument for $q=p^n$. I'm looking for the most elegant and simple way to prove that $p$ still divides $\large q\choose i$. In Java, what is the best way to determine the size of an object? How can we find premade adventures for a virtual tabletop? Me and a couple of friends are just getting into P&amp;P rpgs and are really excited to try something new. However, we have a problem that we live quite far apart from each other. Therefore we'd like to use one of excellent virtual tabletops that are available. But none of us have any real experience with creating an adventure and would thus like to use a premade and beginner friendly adventure. But we're having a hard time finding one with premade story, maps etc in an online format. Most seem to be printed books and the Roll20 marketplace hasn't really gotten started yet. So what we're looking for a is online way to play a premade adventure to get us started in the RPG world.  How can we find these adventures for a VTT? What does var prime = num != 1; mean? How to close view windows? I would like to close some view windows that I opened accidentally.  See the image below:    Edit: another note: I tried searching for the solution, but only found things like "Join Area", which don't seem to work in this situation. Insert eps graphics with graphicx (file not found) Produce two distinct topologies of R such that the first is strictly finer than the second but the two are homeomorphic to each other. Need to prove that they are homeomorphic to each other.  So if T_1 and T_2 are topologies on set X with T_1 contained in T_2, every element of T_1 will be in T_2 and T_2 is the "finer" (stronger and larger) topology. So to them be homeomorphic is T_1 going to have to be a hole in T_2? Why is this command deleting all entities? I'm trying to remove a specific, named, item if it is dropped on the floor. I have the following on my command block:  kill @e[type=Item] {Item:{tag:{display:{Name:"RedFlag"}}}   But it's removing all items, not just the one I want. What am I doing wrong?  I also tried  kill @e[type=Item] {Item:{id:"red_flower",Damage:2}}   But that didn't work either, it's still removing all items, even things that aren't red flowers! Cells not updating automatically I have a formula in a cell which counts the values in 2 other cells. Normally, after entering in the values in these 2 other cells, the formula cell updates automatically when Enter is pressed or when the cells are clicked out of.  However, for some reason, the value calculated by the formula does not update automatically. I am having to click on the formula bar, then I have to click enter for the formula cell to do its thing.  Why is this happening? Rotating a table so that at least two name cards are correct Why is angular velocity of any point about any other point of a rigid body always the same? "R cannot be resolved to a variable"? Drawing different tikz shapes parameterized by data from a file I have a file with three columns of data, x, y, r.  # x y r 1 1 0.1 2 2 0.5 3 3 1.2   and so on. I'd like to loop over the rows, and draw a circle at x,y with radius r for each row.  I first thought I'd use pgfplots with customized marks, but I couldn't figure out how to pass the radius parameter into the pgfplot mark handler. Then I thought I'd use tikz and pgf directly, but I couldn't figure out how to reference the values for r inside a draw command sequence.   Any thoughts ? I'd be very grateful for some help. Proof that $\suṃ̣_{k=1}^{n}k {n \choose k}$ for $n \in \mathbb N$ is equal to $n2^{n-1}$.  As a hint I got that $k {n \choose k} = n {n-1\choose k-1} $.  I tried solving this by induction but, in the inductive step I'm not arriving to the correct result. How to compile and run C/C++ in a Unix console/Mac terminal? How can I compile/run C or C++ in Unix console or a Mac terminal?  (I know it, forget it, and relearn it again. Time to write it down.)