This is the definition used to numerically calculate the maximal Lyapunov exponent: $$ |\delta Z(t)| \approx e^{\lambda t} |\delta Z_0| $$ $$ \lambda \approx \frac{1}{t} \ln\left(\frac{|\delta Z(t)|}{|\delta Z_0|}\right) $$ Consider a system to be bounded in a region of space, like two balls moving inside a room, where we want to see if the dynamics are chaotic or not. We start recording the trajectory of a ball and after a good amount of time, we go back and start from a tiny dislocated initial position. We see that the trajectories diverge and we may conclude that the system is chaotic. Like the following:  But here is what I do not understand. If the room is bounded, then these two trajectories cannot get arbitrarily far from each other and their distance would not be larger than the size of the room. So in the formula for the Lyapunov exponent, $t$ would get larger and larger over time so $\lambda$ would become smaller and finally zero, no matter if it was positive at first. Would this mean the Lyapunov exponent decays with time in bounded systems? A double pendulum is a bounded system, two trajectories (angles) can not be more than $\pi$ apart at any moment, so does its Lyapunov exponent decay to zero over time?
Forgive the possibly naive question, but I'm returning to a basic undergrad-level study of chaotic dynamical systems and can't quite seem to reconcile the notion that in order for a dynamical system to be considered chaotic, it must have positive global Lyapunov exponents. Such exponents, of course, drive the divergence of trajectories. However, topological transitivity implies that trajectories can get arbitrarily close as well over some finite time. It seems that positive Lyapunov exponents would disallow this mixing. Is there a point at which these positive exponents no longer describe the system, and the mixing behavior begins?
Are There Any Mechanics for Removing Possession? So I'm thinking of a few adventure hooks for my players, and I thought about the possibility of them removing a demon that has possessed a noble. Are there currently any mechanics or rules surrounding possession and the removal of entities?  If not, does anyone have any play-tested homebrew rules? Whenever I open a .sh file, it opens it in gedit instead of the terminal. I can't find any option similar to Right Click → Open With → Other Application... → Terminal.  How do I open this file in the terminal? Unable to open Chrome WebDriver in another language I'd really like to write a Selenium WebDriver test that verifies text is localized. The localization code bases the strings it displays on navigator.language (or navigator.userLanguage for IE). I cannot figure out, however, how to launch a Chromium window for a different language.  I see this bug which sort of explains the problem (closed as not a bug):   Here's how I create the browser instance:  var webdriver = require('selenium-webdriver'); var chrome = require("selenium-webdriver/chrome");  var options = new chrome.Options(); options.addArguments('--lang=es'); var caps = options.toCapabilities(); var driver = webdriver.Builder().withCapabilities(caps).build();   I installed the Spanish language, but whereas navigator.languages includes es in its list, but even if I move it to the top of the preferences, en-US is still what navigator.language is set to.  Have any of you found a way to successfully write multi-language tests with Selenium WebDriver? Split a string with delimiters but keep the delimiters in the result in C# RC4 , Is it possible to find the key if we know the plaintext and ciphertext? Is it possible to find the key if we know the plaintext and ciphertext with RC4. How should I write the algorithm? Combining two expressions (Expression>) Checking if two Poisson samples have the same mean How to prevent an email from accidentally being forwarded? What's the difference between fit and fit_transform in scikit-learn models? I'm a newbie to data science, and I do not understand the difference between the fit and fit_transform methods in scikit-learn. Can anybody explain simply why we might need to transform data? What does it mean, fitting a model on training data and transforming to test data? Does it mean, for example, converting categorical variables into numbers in training and transforming the new feature set onto test data? Is my expansion of the state $| x \rangle$ correct? In my quantum mechanics textbook it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by:  $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$  However, i'm not sure how to go about proving this relation. My idea was that the eigenstates of momentum can be written as below:  $\phi(x,p) = \Large \frac{1}{\sqrt{2\pi \hbar}} e^{-ipx/\hbar},$  which form an orthogonal basis.  But, we can expand any state, $|\text{state}\rangle$, in terms of the othorgonal basis right? So is my notation below correct for the expansion of the state $| x \rangle$ ?  $|x\rangle = \int_{-\infty}^{+\infty} \phi(p', x) |p'\rangle dp'$  If it were correct then the following would also be correct?  Using the fact that $\langle p | p' \rangle = \delta(p - p')$, $\langle p | x \rangle$ can be computed:  \begin{align} \langle p | x \rangle =&amp; \int_{-\infty}^{+\infty} \phi(p', x) \langle p | p' \rangle dp' \\ =&amp; \int_{-\infty}^{+\infty} \phi(p', x) \delta(p - p') dp' \\ =&amp; \phi(p, x) \\ =&amp; \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \end{align} as required.  Does the logic work here? I know that contents of double quotes are expanded, whereas the contents of single quotes are not, such that  echo '$1'   gives  $1   where as   echo "$1"   gives  &lt;the argument - blank in this particular example&gt;   the same as   echo $1   However, the question  led me to wonder why double quotes and not single quotes were used when declaring the alias, thus:  alias cutf="_cutf"   instead of   alias cutf='_cutf'   I would have employed single quotes as in the latter example. Most examples on the web use single too, unless there is some expansion required. However, in this case, no expansion, to my eyes, is apparent. Is it that, in this case, they are interchangeable, or are the double quotes necessary because a function definition is employed? How to get access to USB-storage from an application installed as Snap? The answer to tactical downvoting problem? As we know, the tactical downvoting (I'm talking especially about SO, as I don't know about other sites) is more and more becoming a serious problem, as I see, the community can't stop this behavior without some sacrificing.  As a suggestion, I think it makes sense to prevent an answerer from downvoting any answer on the same question, this -sure- has a harm, but IMO, it's less harmful than the tactical downvoting issue, and to make the things a bit more easier, we can prevent an answerer from downvoting a question without posting a comment to that answer before.  And vice versa, if a user downvoted an answer, he shouldn't be able to post another one unless he posted a comment first, to prevent working-around that system, so a user can't downvote other answers before posting his own to avoid posting comments.  EDIT, more clarification: I don't say wrong answers shouldn't be downvoted, BUT if you want to downvote an answer and post your own, you'll have to post a comment, so we make sure you're not just downvoting right answers tactically. "batch" tag turns into "batch-file" I installed jockey-gtk but how do I run it? zuber@zuber-pc:~$ sudo apt-get install jockey-gtk   Reading package lists... Done Building dependency tree        Reading state information... Done The following packages were automatically installed and are no longer required:   nvidia-common python-xkit Use 'apt-get autoremove' to remove them. The following NEW packages will be installed:   jockey-gtk 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/3,158 B of archives. After this operation, 53.2 kB of additional disk space will be used. Selecting previously unselected package jockey-gtk. (Reading database ... 156414 files and directories currently installed.) Unpacking jockey-gtk (from .../jockey-gtk_0.9.7-0ubuntu11_all.deb) ... Setting up jockey-gtk (0.9.7-0ubuntu11) ... zuber@zuber-pc:~$ jockey-gtk The program 'jockey-gtk' is currently not installed. You can install it by typing: sudo apt-get install jockey-gtk   And this not only occur with jockey-gtk. I have same problem with synergy(I did not tried with other software). Is it known problem or am I only one? How do I fix it? Indefinite integral of secant cubed I need to calculate the following indefinite integral:  $$I=\int \frac{1}{\cos^3(x)}dx$$  I know what the result is (from Mathematica):  $$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$  but I don't know how to integrate it myself. I have been trying some substitutions to no avail.   Equivalently, I need to know how to compute:  $$I=\int \sqrt{1+z^2}dz$$  which follows after making the change of variables $z=\tan x$. When I was a kid we had a VHS tape of some Disney live action (not animated) sci-fi movie. It may have been a mini-series or a made-for-tv movie, but I'm not sure. I'm also not sure about the year but it would probably have been late 80s or early 90s based on when I would have seen it. I know it was Disney because they had some live action Goofy and Mickey running around the movie's set as an introduction discussing the plot with a host before the movie.  The plot is basically as follows:   A starship is loaded with mostly children and teenagers as a cross-galaxy human settling effort. Accidents (which are not really accidents and later revealed) cause adult crew members to be killed.  They travel to another planet and face all sorts of troubles. It arrives at the new planet after overcoming obstacles.   Other memorable scenes that may help:   Navigating out of earth's asteroid belt made out of space junk. One of the adults was killed via an airlock malfunction. The ship was supposed to be peaceful with no weapons. At one point they bring on board some crazy guy who puts together a rail gun. There's a memorable line like "Do you want no weapons at all or a well aimed rock?"  There's a planet or space station (I can't remember which) that has a fight scene with the evil bad guy. This scene takes place on what looks to be a carnival or fairground. The bad guys are trying to steal the ship the kids are on because it has some kind of really good futuristic space drive.   Hopefully you remember it better than I do! I've tried looking online for it before with no luck. Can't watch youtube videos with chromium on ubuntu 13.04 64-bit I've digged all over google but I can't find any solution to my problem.   Firefox works pretty well. I have this issue only with chromium and only with youtube. I have no problem watching videos at any other site with chromium.   When I try to watch a video a black screen occurs and after a while it gives me the warning "an error occurred". Weirdly, sometimes when I try a couple of refreshing it starts to play but then the issue occurs again.  I tried removing-reinstalling chromium or the flash plugin but it did not do any good. How to return to original .pdf presentation after open a .pdf linked file? I'm preparing a beamer presentation and I'm using hyperlinks for the first time; When I do something like:  \href{name_file_to_open.pdf}{here I put a link to a figure which is working well}   It works just fine and open the .pdf document; Problem is I can't return to my presentation, cause when I close the linked.pdf I close it all. (I try this with movie and indicating an area to the file open like \href[width=3cm,height=2cm] but then it doesn't work)  Can anyone please has any suggestion to this problem? Is "Michael here" correct?