Uniform convergence of $f_n(x) = \frac{nx}{1+n^2x^2}$ on $[a,\infty), a>0$
Uniform convergence on two separate intervals
How to avoid becoming stressed while playing endurance games? Although I am asking for Super Hexagon, I've noticed there is a multitude of games (rhythm games in particular) where the more stressed I am, the worse I play and/or are more likely to make a mistake. Often then the games are designed in such a way to induce stress, such as various ways of telling me how far I am, which also is telling me just how much it would suck if I were to fail right now.  This leads to a stress cycle, where I am noticing I am stressed, and I am aware of this being a bad thing, which makes me more stressed, over and around until I eventually fail.  This often means that I probably could've done &lt;some new level I just unlocked&gt; just fine, except I saw that it was new, got stressed about it and started the stress cycle.  So how can I:   Avoid getting stressed while playing Defuse the stress cycle once it has inevitably started   One thought I had for this is simply taking deep breaths, but particularly in Super Hexagon, there is no way to pause and taking deep breaths is distracting enough to make me fail. I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...) = 1 - S$$ Therefore, $2S = 1$, so $S = \frac{1}{2}$. QED   I was under the impression that this sum doesn't converge, and therefore this step of the proof is invalid:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$   That is, it's not even valid to suppose that the sum is equal to some (implicitly real or complex) number $S$, and therefore the reasoning that follows is meaningless because $S$ doesn't exist and therefore has any properties we want it to have.  Is the reasoning I've given above correct? That is, is it correct for me to claim that the proof fails because $S$ doesn't exist?  (I've heard that there are techniques for evaluating divergent integrals by using complex analysis and Taylor series, so perhaps there's another way to prove that the summation is $\frac{1}{2}$; I just wanted to see whether my reasoning is sufficient to explain why this particular proof is incorrect. If the proof actually is correct, then I stand corrected!)  Thanks! Can I get more than 1000 records from a DirectorySearcher? I just noticed that the return list for results is limited to 1000. I have more than 1000 groups in my domain (HUGE domain). How can I get more than 1000 records? Can I start at a later record? Can I cut it up into multiple searches?  Here is my query:  DirectoryEntry dirEnt = new DirectoryEntry("LDAP://dhuba1kwtn004"); string[] loadProps = new string[] { "cn", "samaccountname", "name", "distinguishedname" }; DirectorySearcher srch = new DirectorySearcher(dirEnt, "(objectClass=Group)", loadProps); var results = srch.FindAll();   I have tried to set srch.SizeLimit = 2000;, but that doesn't seem to work. Any ideas? I want to make a JavaScript application that's not open source, and thus I wish to learn how to can obfuscate my JS code? Is this possible? Light emission from the vicinity of a black hole The photograph of the black hole, at the centre of galaxy M87, is now a part of human history. We are told that the red-colour is caused by hot gas, matter &amp; radiation around the "edge", the Event Horizon. Given the power of the gravitational forces emanating from black hole, why aren't these three things drawn into the abyss? If light is bent/ drawn into the BH, how does this red image arise? The logic, here, would be that a body, on the Event Horizon, is immune from the gravity--which sounds pretty fatuous? Projection on the hyperplane $H: \sum x_i=0$ Consider the hyperplane $H: \sum_{i=1}^n x_i=0$. Let $P$ denote the orthogonal projection onto $H$. What exactly is the matrix $P$ and how do we find it? How do I see a C/C++ source file after preprocessing in Visual Studio? Probability density and distribution function I am new to probability course and find concepts very confusing. I am learning maximum likelihood estimation and as a starting point for that do we find the density or the distribution?   In general, do we find the distribution from density or is it vice versa and how does one find it? For example, for Gaussian distribution do we find the density of a variable or the distribution so as to estimate unknown parameter A in the case x[n] = A + w[n] where w[n] are uncorrelated gaussian noise. This may sound too trivial but I am really confused. In many contexts I have come across where pdf = prob. density function and in some areas I find pdf = prob distribution function !! Mechanism of Screenshot Basically I with my team are figuring out a way to block screenshots taken on our corporation's computers (Mac mostly). I have searched it extensively and found that there is something called Screen buffer which is being accessed by any screenshot taking software (eg. Snipping Tool etc). I tried to find the exact mechanism and the actual implementation of such softwares but with no luck. Also I found out that Netflix uses some kind of technology which corrupts screenshot image and blackens it. I did not find any actual mechanism or implementation to study it further. I also came across something called Hardware overlays but did not understand it properly in order to use it for actual implementation. I have considered a way to edit scancode maps in registry but I am looking for a more general way. I read a few articles for apple's FPS though. Can somebody please guide me through this? Some good resources for the same would be highly appreciated. Any particular or relevant forums or subs to discuss this are also welcome Local Coordinate to Geocentric The ultimate question, I need a transformation matrix to take a point in local space representing a roughly 500m x 500m place in New Mexico centered at -108.619987456 long 36.234600064 lat.  The final output needs to be in geocentric coordinates and I can during initialization get any number of points on the map such as the corners, center, western most along the center etc.  During run time I would like to have a transformation matrix that can be applied to a position in local space expressed in meters and get back an approximate GCC coordinate.  The center of the point in GCC: -1645380 -4885138 3752889 The bottom left corner of the map in GCC: -1644552, -4881054, 3749220  During run time I need to be able to multiply (-250, -250, 0) with the transformation matrix and get something close to (-1644552, -4881054, 3749220).  I've spent more than a week trying to research a solution for this and so far nothing.  Given an identity matrix as our starting position and orientation. Then using geotrans and the known lat, long, and height of the starting position I get an x,y,z geocentric coordinate. A vector from the origin of (0,0,0) gives both the up and translation for the matrix. However, I need the forward and right so that I can pass a distance in meters from the origin into the transformation matrix and get a roughly accurate GCC. Do I have all of the inputs I need to calculate the right and forward?  Inputs Origin: 0, 0, 0 Global: -1645380, -4885138, 3752889 Up (Normalized): Global - Origin  Desired Outputs Right: ? ? ? Forward: ? ? ?  So with the right and forward added to the up and translation I already calculated I would have a transformation matrix.  I could then apply the matrix to a vector of say (50, 50, 0) and get something within about 0-3 cm of the output if I feed the lat/long back into geotrans.  This matrix would only ever be used small maps of about 500m x 500m so the curvature of the earth is negligible.  In reply to whuber,  I don't know your level of experience so I will start with the very basics.  An identity matrix is a 4x4 matrix that you can multiply by a vector and it returns the vector unchanged. Such as below.  1 0 0 x=0 0 1 0 y=0 0 0 1 z=0 0 0 0 1  The x, y, and z of the matrix are how much to translate the vector by and the first 3 rows and columns are the rotation.  Below is a tranformation matrix that does nothing to the orientation of a vector but does translate.  This is what you would want for two axis aligned worlds in different locations.  1 0 0 13 0 1 0 50 0 0 1 -7 0 0 0 1  If you were to multiply a vector of (10, 10, 10) with the above transformation matrix you would get an output of (23, 60, 3).  My problem is the axes are not aligned and the "plane" of the world I am trying to get the local coordinate converted to is projected on the ellipsoid of the earth.  Geotrans is library that you can use to pass a coordinate from one system such as geodetic or GDC (lat, long, height) and get back another such as geocentric or GCC (x,y,z).  For example: If my game world was representing a map centered exactly on the prime meridian and equator the transformation matrix would look something like below. I am just guesstimating the Y and Z rotations and might have them flipped and/or the wrong sign.    0 0 1 6378137 0 1 0 0 1 0 0 0 0 0 0 1 Is there any way to automatically get notified of property changes in a class without having to write OnPropertyChanged in every setter?  (I have hundreds of properties that I want to know if they have changed).    Anton suggests .  I've actually used the "Castle" library for something similar in the past, and while it does reduce the amount of code I've had to write, it added around 30 seconds to my program startup time (ymmv) - because it's a runtime solution.  I'm wondering if there is a compile time solution, maybe using compile-time attributes...    Slashene and TcKs give suggestions which generates repetitive code - unfortunately, not all my properties are a simple case of m_Value = value - lots of them have custom code in the setters, so cookie-cutter code from snippets and xml aren't really feasible for my project either. How do I tell if matrices are similar? Javascript Date: next month How much memory can a 32 bit process access on a 64 bit operating system? In English, especially in a poetic description, ships and cars are  referred to as "she".   Maybe their owners compare their beauty and elegance with those of women, but what if a female owner wanted to describe her yacht or car?   Would she also say, "She is a real beauty", or it might be "he" then?   Does the choice of a pronoun depend on whom she is giving the description of it to — a man or a woman?  What about a man-o’-war or a merchantman? Being ships, are they also referred as "she"?  P.S.  on the use of "she" in reference to ships, but there seems to be no answer to my question there.  P.P.S. My question wasn't pulled out of a thin air. I came across a book by a prominent Russian linguist, G.Veichman, written in 1990, (Novelties in The English Grammar is my translation of the title), where I read that the rule of personification of vehicles may be added regarding a female owner talking about her car using the preposition "he". Period. No further explanation or reference examples. Compute $\int_{0}^{\infty}\frac{x\sin 2x}{9+x^{2}} \, dx$ I'm a committer for the .  It went to the private beta phase sometime in the last 12 hours, but yet to have been invited. When I visit  I get the standard "To log in, you must have commited to the Area 51 site proposal and received the invitation email." message.  However, when I navigate to the Pets site via the Android app, I'm able to view the site in it's entirety (questions, answers and comments). When I attempt to post, I get a message saying my user token isn't allowed to take action on this site.  Same behavior is observed when using the iOS app. What I want is to create a bootable pen drive that shows GRUB menu and I can select any of the several bootable systems from there to install/run. I have several ISO images that I want to install in the USB like Ubuntu, Fedora, System Rescue CD, Windows XP and Windows 7 (The windows versions are optional, mostly am looking for a multi installer for Linux distros, specially Ubuntu version)  The default Startup Disk Creator only works with Linux Images and only 1 at a time (At least for the mean time. If it allows more systems and multiple ISOs it would be awesome)  What method exists that I can use step by step to create a bootable USB Drive with multiple images than can be booted from the USB and selected from GRUB. It must support Windows Images and Linux Images.  Alternatively, is there also a way to have multiple versions of the Ubuntu installation ISO on same USB Drive, such as, when booting from it, I get to choose what Ubuntu version and architecture to install, for example:     BOOT MENU   Ubuntu 12.04 32 Bit   Ubuntu 12.04 64 Bit   Ubuntu 12.10 32 Bit   Ubuntu 12.10 64 Bit   UPDATE: Wanted to add that the English version for the post mentioned by daithib8 is here: How do I add a Pokestop to an area that has none? It is an area where there are tons of businesses but unless I drive about a mile from my work area there are no Pokestops.   I want to add one since we have 1000's of people in the work area, a lot of them play Pokemon Go, and we all can't leave work to drive a mile away to get to a Pokestop area.   Plus it would be nice to be able to capture Pokemon on our breaks. Now, that Ubuntu 17.10 is out, and Unity 7 is replaced with Gnome 3, is there a way to install and use Unity7?   I know Unity is still in the repositories, so it should be possible, right?