Convertion of Floating Value to Int 0.345 to 345 in java language
How do I generate random integers within a specific range in Java?
This is the proof from the book:     Theorem. There are infinitely many primes of the form $4n+3$.    Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in this form.  Proof of Theorem: Let assume that there are only a finite number of primes of the form $4n + 3$, say  $$p_0, p_1, p_2, \ldots, p_r.$$ Let $$Q = 4p_1p_2p_3\cdots p_r + 3.$$ Then there is at least one prime in the factorization of $Q$ of the form $4n + 3$. Otherwise, all of these primes would be of the form $4n + 1$, and by the Lemma above, this would imply that $Q$ would also be of this form, which is a contradiction. However, none of the prime $p_0, p_1,\ldots, p_n$ divides $Q$. The prime $3$ does not divide $Q$, for if $3|Q$  then $$3|(Q-3) = 4p_1p_2p_3\cdots p_r,$$ which is a contradiction. Likewise, none of the primes $p_j$ can divides $Q$, because $p_j | Q$ implies $p_j | ( Q - 4p_1p_2\cdots p_r ) = 3$, which is absurd. Hence, there are infinitely many primes of the form $4n +3$.   END  From "however, none of the prime ...." to the end, I totally lost!    My questions:   Is the author assuming $Q$ is prime or is not? Why none of the primes $p_0, p_1,\ldots, p_r$ divide $Q$? Based on what argument?     Can anyone share me a better proof?     Thanks. A groupoid is defined to be a category where every morphism is an isomorphism. So sometimes a group is said to just be a groupoid with one object.   When I try to make sense of this, I denote the single object as $G$. I view the morphisms as the analogue of "elements." The identity $1_G$ is the analogue of the usual identity $e$, we can compose any two morphisms since they are all arrows on $G$, and for any arrow $f$, we have some $f^{-1}$ such that $f\circ f^{-1}=f^{-1}\circ f=1_G$, so the idea of inverses is still there.  So I informally associate the elements of the group in the usual definition to be the arrows in the groupoid. But what does the sole object $G$ in the groupoid "correspond to" if I were to try to informally make sense of a group in the usual sense? Does it even correspond to anything? Distribution of Dark Matter around galaxies It is well-known that measurements of the velocity profile in galaxies are not compatible with Newtonian laws. A way to circumvent the problem is to assume that galaxies are surrounded by a spherical halo of Dark Matter. The density of mass of this Dark Matter can then be computed from the velocity profile with Newton's laws.   My question is the following: the formation of a galaxy involves essentially only the gravitational interaction. According to Newton (and General Relativity too), the trajectory of a particle does not depend on its mass. As a consequence, I do not understand how usual matter and Dark Matter could have followed different trajectories leading to such different mass distributions nowadays. Can somebody explain this (apparent) paradox ? Hubble's law says that the universe is expanding. How come the Milky Way and the Andromeda galaxies are on a collision course? How will they end up colliding with each other? My MATLAB launcher doesn't work I've installed MATLAB R2011a on Ubuntu 11.10 today following . I've also followed the steps to create the launcher. It appeared, but when I click on it nothing happens. Does anyone knows what can I do to fix it? Additional info I can run it perfectly through terminal. My only problem is getting the launcher to work. To create the launcher I used :   Get an icon:  sudo wget http://upload.wikimedia.org/wikipedia/commons/2/21/Matlab_Logo.png -O /usr/share/icons/matlab.png   Get the launcher file:  sudo wget 'https://help.ubuntu.com/community/MATLAB?action=AttachFile&amp;do=get&amp;target=matlab-r2011a.desktop' -O /usr/share/applications/matlab.desktop The proof of shrinking coefficients using ridge regression through "spectral decomposition" I have understood how ridge regression shrinks coefficients towards zero geometrically. Moreover I know how to prove that in the special "Orthonormal Case," but I am confused how that works in the general case via "Spectral decomposition." What does "Warning: untrusted X11 forwarding setup failed: xauth key data not generated" mean when ssh'ing with -X? Currently I'm running Linux-3.0 and I want to update it to Linux-3.3.1 the latest stable kernel release as mentioned at . Can I update to 3.3.1 in Ubuntu without any risk of crashes? I'm updating my kernel regularly as provided by the Update Manager..Currently I have Linux-3.0.0.17..Can I update?  NOTE:  The latest kernel version any user is supposed to use in Ubuntu is updated automatically via the Update Manager, so no action is normally required by user regarding kernel upgrades. What the question author is referring to is mainline kernel, see: How do I run .sh scripts? Is there a way to programmatically determine if a font file has a specific Unicode Glyph? I'm working on a project that generates PDFs that can contain fairly complex math and science formulas.  The text is rendered in Times New Roman, which has pretty good Unicode coverage, but not complete. We have a system in place to swap in a more Unicode complete font for code points that don't have a glyph in TNR (like most of the "stranger" math symbols,) but I can't seem to find a way to query the *.ttf file to see if a given glyph is present.  So far, I've just hard-coded a lookup table of which code points are present, but I'd much prefer an automatic solution.  I'm using VB.Net in a web system under ASP.net, but solutions in any programming language/environment would be appreciated.  Edit: The win32 solution looks excellent, but the specific case I'm trying to solve is in an ASP.Net web system.  Is there a way to do this without including the windows API DLLs into my web site? Do we need a global test before post hoc tests? Examples in which the light maximizes the optical path length I posted a similar  about geodesics on Math.SE. Many sources ( for instance) claim that the light could maximize the optical path length in some cases. But I don't think it's actually true since between two points I can always imagine a path with arbitrary length. Therefore, or the light minimizes the optical length or it's stationary (neither maximum nor minimum). My question is: is the trajectory in red in the figure below a local minimum or a stationary curve?   If I'm wrong, please provide some examples which the light propagates in trajectory of maximum length. I've been using the == operator in my program to compare all my strings so far. However, I ran into a bug, changed one of them into .equals() instead, and it fixed the bug.  Is == bad? When should it and should it not be used? What's the difference? Invariance of a maximally entangled state under unitary operation $U \otimes U^\dagger$ Declare a const array Logging in as the same user multiple times in Remote Desktop Windows Server 2008 Quick question: I have a situation where I need to let multiple people on different PCs log into one server 2008 machine as administrator simultaneously over remote desktop. I have the CALs for it, it's just not set up correctly. When one user tries to log in, it boots the other out. What I need is to present to them a different session, just each as logged in as admin. Sorry for the slightly rambling post, I'm new here. Thanks! Do Stack Overflow and other Stack Exchange sites have an app I can use? What is $\Delta t$ in the time-energy uncertainty principle? I want to copy the data folders from a previous ubuntu(14.04) partition My hard drive is a mess and I want to reformat it and do a fresh install of 16.04. My problem is I only have access to the HDD via 16.04 on a USB stick and although I can navigate to the various folders, nothing I do allows me to copy to an external hard drive. The folders are not encrypted. I also have a small swap partition - I would like to preserve the data on that too if possible.  Here's a listing of the home directories on the drive:  ubuntu@ubuntu:/$ ls -l /media/*/*/home /media/ubuntu/0e228f65-dac7-435f-828b-1eb1a6c3f29a/home: total 4 drwxr-xr-x 46 1000 1000 4096 Dec 21  2015 simon  /media/ubuntu/db79a6d1-7c6c-4d8c-822b-ba6ccd1ddecd/home: total 4 drwxr-xr-x 27 1000 1000 4096 Apr 17  2014 simon  /media/ubuntu/ebe2a6ba-9cb5-4279-8352-5082b8dd4d5e/home: total 4 drwxr-xr-x 39 1000 1000 4096 Aug  3 14:21 simon Incremental (delta) backup of a encrypted data I have several files that I need to keep encrypted on my home hard disk.  I currently use a TrueCrypt file container, currently around 100MB in size. I need to backup this file daily, and I'm looking for an incremental backup solution.  I read that on encrypted volumes even a small change in a file would change the whole volume and prevent the use of an efficient incremental backup solution. Any ideas how to backup in this case in an efficient way which would allow incremental solution?  I currently use TrueCrypt as my backup solution and CrashPlan as my backup utility, but I'm willing to experiment with alternatives.