Specify a bijection from [0,1] to (0,1].
How to define a bijection between $(0,1)$ and $(0,1]$?
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? Trying to remember a book about a human male travelling to alien planet Limits of Poisson's ratio in isotropic solid Where does the "apply bonuses and penalties in the most beneficial order" rule come from, if it exists? I have seen quoted several places on the internet a rule that one can apply bonuses and penalties in the most beneficial order.  An example I come across frequently (as a druid) is the combination of Natural Bond and . Natural Bond increases your effective druid level for determining the stats of animal companions but cannot increase it beyond your character level. Choosing an alternative animal companion lowers your effective druid level for determining the same stats. The "most beneficial order" rule is used to justify applying the level penalty before the natural bond bonus in order to get around the character level cap on natural bond.  Does this rule actually exist? If so, where does this rule originate? I would like to be able to cite a reliable source for my GM. How can i change formatting of the table of contents to avoid section numbering and section title to overlap?  Document:  \documentclass[10pt,oneside,notitlepage]{book}  % Use Heuristica Font \usepackage{heuristica} \usepackage[heuristica,vvarbb,bigdelims]{newtxmath} \renewcommand*\oldstylenums[1]{\textosf{#1}}   \begin{document}  \tableofcontents  \part{Part one}  \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh}   \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi}  \end{document}   Example output: I read a very special book a few years back. We follow a boy who lives in a world where, if one moves far enough in a direction, that person gets distended and changed in size.   The boy works on laying tracks for the train, on which his city is. The city has to continually keep moving or it also will get squashed.   It's like a world like if one end is pinched and then dragged out like a piece of rubber string or bubble gum.   The book is not older than 10 years I think.  Hope I was clear enough... Find asymptotic of recurrence sequence Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$.  The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$.  I tried to find explicit formula for $x_n$, but did not succeed. I stucked at that point: if $a_1=(\frac{1}{4})^2$, $a_n=(a_{n-1}+\frac{1}{4})^2$, then $x_n=-a_{n-2}+\frac{1}{4}$ for $n\ge3$.   Thanks for any ideas and help. If you have a conducting hollow sphere with a uniform charge on its surface, then will the electric field at every point inside the shell be 0. The reason the electric field is 0 at the center is clear from the symmetry of the sphere, but for a point at a certain distance from the center, shouldn't a net electric field exist?  I saw a similar question on the site but it was vaguely explained. Please require more reviewers on First Posts On the one hand, I see very few posts in the review queue when I visit.  On the other hand, I see a lot of pretty awful first posts sitting on the front page.  This leads me to suspect that robo-reviewers are blowing through the queue.  Given how empty the queue is, could you please increase the number of people who have to check in before a post leaves the queue?  Of course, some honey-pot solution to detect and disable the auto-clickers would be even better. What is the strict aliasing rule? How many rectangles or triangles. I have come across numerous questions where I am given the picture such as the above one been asked "how many rectangles are there?". I have even come across some slightly different images that instead of rectangles you are supposed to find the number of triangles. Well, I was thinking whether there is any formula or strategy that is used to solve these problems without having to manually count every shape.  Help would be appreciated.  Thank you :) What strategies, criteria, or rules to use for selecting coordinate systems? What strategies, criteria, or rules do you use for selecting coordinate systems for    (a) storing,  (b) analyzing, and  (c) displaying GIS data?     (I humbly offer  to a  about watershed analysis as an example of the considerations involved in (b).)  What are the pitfalls to watch out for?  Links to Web sites you find particularly helpful in this regard are welcome. That's really embarrassing, however I need to ask it. I could not prove that the normed space    $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle $ is complete (as a metric space), where $\| f\|_1 = \int_0^1 |f(x)|dx$. I don't really know how to explicit a continuous functions as a limit of a Cauchy sequence. I tried to prove that the pointwise limit (of the Cauchy sequence) have just a finite number of discontinuities (I don't know if it's true) and then approximate it by a continuous function.  Thanks in advance. Apparently, you can speed up your computer by adding more RAM. Is the amount of ram to be added is directly or indirectly related to the amount of hard-disk  space, processor speed, Motherboard etc... Can the memory of a RAM be compared with Virtual Memory and Cache Memory?  Note: About my PC Configuration: Intel core i5 processor, Motherboard DP55WB 4GB RAM, 700 GB Hard-disk space. I have 16GB of system RAM. On startup with no applications open except the task-manager Windows is using about 3gb of RAM. I looked in the processes tab, but nothing seems to be out of the ordinary. How can I find out why my Windows is using so much RAM.    all processes from all users      Reading from the poolmon it seems that my wireless broadcom driver is using about 0.4GB of of RAM. Even if I remove  it would still be using  2.6GB on start-up, which is still too much.      After reinstalling the wireless driver associated with the memory leak. I have a new screenshot and would like to confirm that it was indeed a memory leak. Are nuclear weapons simply non-existent in the Star Wars universe? The resistance is attacks the Starkiller Base with what appears to be pretty small time ammunition even by modern earth missiles, which requires a long running sequence of repeated flybys.  The objective, IIRC was to destroy a critical piece of machinery on the surface which would then destabilize the entire planet/weapon (feel free to correct in edit if I remember incorrectly).  OK cool, but then why the force wouldn't they just drop a nuclear bomb on it?  Have nuclear weapons been used any other time in Star Wars (including EU), or, like smart phones, have they just not been invented in that world? Using Process.Start() to start a process as a different user from within a Windows Service I'd like to periodically run an arbitrary .NET exe under a specified user account from a Windows Service.   So far I've got my windows service running with logic to decide what the target process is, and when to run it. The target process is started in the following manner:   The Windows Service is started using "administrator" credentials. When the time comes, an intermediate .NET process is executed with arguments detailing which process should be started (filename, username, domain, password). This process creates a new System.Diagnostics.Process, associates a ProcessStartInfo object filled with the arguments passed to it, and then calls Start() on the process object.   The first time this happens, the target process executes fine and then closes normally. Every subsequent time however, as soon as the target process is started it throws the error "Application failed to initalize properly (0xc0000142)". Restarting the Windows Service will allow the process to run successfully once again (for the first execution).  Naturally, the goal is to have target process execute successfully every time.  Regarding step 2 above: To run a process as a different user .NET calls the win32 function CreateProcessWithLogonW. This function requires a window handle to log the specified user in. Since the Windows Service isn't running in Interactive Mode it has no window handle. This intermediate process solves the issue, as it has a window handle which can be passed to the target process.  Please, no suggestions of using psexec or the windows task planner. I've accepted my lot in life, and that includes solving the problem in the manner stated above. The  lists various combinatorial objects that are described by them. I posit that there might be bijections between these various combinatorial objects. For some of them (like Dyck paths, correctly matched parentheses and paths from the bottom left to top right of a $2n \times 2n$ grid) they are quite obvious.   I was then trying to find a bijection between the number of Dyck words and the number of ways of associating $n$ applications of a binary operator to $n+1$ items (third one on the list). I attempted to do this for a simple case ($n=3$ which is the example provided in the Wikipedia article). However, couldn't find one after multiple hours. Is it reasonable to expect such a bijection will exist? If so, how do we go about finding it?    EDIT: In addition to the very nice answer by @Marc, the following page also helped me see the bijection:   "Let $P$ be the Dyck path and $f(P)$ be the binary tree. If you go up in the Dyck path, create a left child. Otherwise, go up one vertex until creating a new right child is possible and create one."    Here is one of my attempts:   Number of Dyck words with length $2 \times 3$ is $\frac{6 \choose 3}{4} = 5$. They are:  hhhttt; hhthtt; hhttht; hthhtt; hththt  And the number of applications of a binary operator among $3+1=4$ factors is:  ((ab)c)d;     (a(bc))d;     (ab)(cd);     a((bc)d);     a(b(cd))  Both combinatorial objects have been arranged in a way that there is some kind of ordering between them. For the Dyck words for example, if h equals +1 and t equals -1, then the order is lexicographical from the left to the right of the cumulative score along the sequence.   Now, the first and last characters of the Dyck words are always h and t respectively. So, we can ignore them. We are left with:  hhtt; htht; htth; thht; thth  I tried to start from the left of the sequence abcd and if I see 'h', merge the character with the one to its right. This approach didn't produce a valid mapping from the third Dyck word to the third binary operator precedence order. How can I install a Nautilus script? OSGi/Felix Declarative Services: How to filter the services to be bound? I am using Apache Felix and its Declarative Services (SCR) to wire the service dependencies between bundles.  For example, if I need access to a java.util.Dictionary I can say the following to have SCR provide one:  /** * @scr.reference name=properties interface=java.util.Dictionary */  protected void bindProperties(Dictionary d) { }  protected void unbindProperties(Dictionary d) { }   Now, I have more than one Dictionary service available, and I want to filter them using the "name" service property (I only want "name=myDictionary"). I can do that with code (using a ServiceTracker), but I'd rather specify the filter in the @scr annotation instead.