Is the universe infinite?
Is the universe finite or infinite?
Can I install the 64 bit version? How can I obtain a Cauchy distribution from two standard normal distributions? I am interested in      Let $X\sim N(0,1), Y \sim N(0,1)$ independently. Show $\frac{X}{X+Y}$   is a Cauchy random variable.   My work:  $f_{X,Y}(x,y)=\frac{1}{2\pi} e^{\frac{-1}{2}(x^2+y^2)}, -\infty&lt;x,y&lt;\infty$ by independence  Let $U=\frac{X}{X+Y},V=X+Y$. (Is there a better $V$ to choose for this bivariate transformation?)  Then, $X=UV, Y=V - UV$. So, $|J|=V$.  $f_{U,V}(u,v)=f_{X,Y}(uv,v-uv)|J|=\frac{v}{2\pi}e^{\frac{-v^2}{2}(2u^2-2u+1)},-\infty&lt;u&lt;\infty$  $f_U(u)=\frac{1}{2\pi}\int_{-\infty}^{\infty}ve^{\frac{-v^2}{2}(2u^2-2u+1)}dv$. Let $y=\frac{v^2}{2}(2u^2-2u+1)$, so $dy=v(1+2u^2-2u)dv$. Then,  $f_U(u)=\frac{1}{2\pi(2u^2-2u+1)}\int_0^{\infty}e^{-y}dy=(\pi[(\frac{u-1/2}{2})^2+1])^{-1}, -\infty&lt;u&lt;\infty$,   which is not exactly a Cauchy distribution. Where did I mess up? More importantly, how would you proceed in solving this problem? Is this asbestos? (South korea) School in South Korea removed asbestos over the past few weeks. Now I am all paranoid.   Is this asbestos? Found these dime-sized chunks by my desk.          Higher resolution images are available here:    There should be 4 pics. If not clear enough let me know. How do I export and import content types (including fields) between environments? Arduino Negative Voltage Reading with Positive Ground I am trying to measure voltage &amp; current off a solar charge controller from panel, battery &amp; load simultaneously with my arduino. I have managed to achieve this by tying all the negatives together and using it as GND, however, when I do that the charge controller doesn't function properly and when I switch to off-mode the loads still run. After checking the instruction manual further, I found out that the positives are shared common between Pan, Bat, Load. This was confirmed with a multimeter.   So now what I am thinking of doing is using the positive from the battery as the GND reference and then measuring all other voltages (through voltage dividers with respect to the GND). The voltages should then range between -17V to 0V.   However, I have read that the analog input pins are not able to read negative voltages, and with respect to the positive GND, the Analog inputs will all be negative.   EDIT: I was thinking of using an amplifier to invert positive ground to a negative ground, meaning I could measure all the positive sides of the voltages. But after getting answers from you guys it seems I may not have to use an amplifier and a voltage divider may suffice. I have currently included a 2 resistor voltage divider between each analog input, +5V source, Vmeasure which you can see in the diagram below.  According to my calculations, if this works, @-17V I should see 0.6V @ analog input. @0V I should see 4V at analog input.   I'd appreciate feedback on this circuit, and below in the second image I've also included a piece of another method suggested by @jonk (thanks), and would be interested in knowing what the benefits are of method B) over method A)  Thanks!  Ps. the ameters are there in place of ACS712 current sensors which also go into analog input pins.   Method A)        &ndash; Schematic created using   Method B (EDIT: This method has worked, I am still yet to test method A) thanks @Jonk)) What is the population of the Star Wars Galaxy? Powershell auto complete settings. I was wondering if there was anyway to change the way the current powershell autocomplete works. I have been using bash and had changed them to offer a few features that I'd like in my powershell experience.   Currently powershell as I'm writing a command with a folder it will tab autocomplete and also cycle through my folders. This works how I like but one feature that I miss is using partial matching of a command and hitting the up key and it would find that command that matches the rest of what I do.   So Id write 'echo test'  Then id write other commands and try 'ec' and tap up and 'echo test' would complete and not the last command I typed  Just wondering if there is anyway anyway to get this back? php Replacing multiple spaces with a single space Android reading from .txt File created in src folder I am new to linux/ ubuntu and just downloaded the v11 of ubuntu to learn linux. I faced this "No root file system is defined" error while trying to install it. This is how I was proceeding -    Downloaded ubuntu v11 from the ubuntu.com website, wrote the iso into a bootable cd, and then got into the linux installation process on restarting; everything fine till here. In one of the screen during installation, there are three choices - install alongside windows, something else etc.. Since I have windows 7 in the C drive, I selected the first option - "Install alongside another OS".      Let me interrupt here to give my existing system details - HardDisk   size: 640GB.  Number of existing partitions - 5 C drive - label   windows - 160GB - has windows 7 installed D drive - label linux -   160GB - does not have any data E drive - label backup - 100GB - has   some data F drive - label entertainment - 100GB  - has some data G   drive - label work - 90GB  - has some data All of the above are in   NTFS format.    In the screen that follows, an image is shown with 2 parts and each part is shown to have size equal to 80GB. There is a horizontal slider in the middle of these two images, with which I can resize the partitions. The right side partition is labeled Linux. There is also an option to do 'advanced partitions'. On clicking advanced partitions on the above screen, a new screen is shown in which the partitions existing in my computer are shown - but the labels are different. I see that there are two partitions with size = 160GB each (as expected), and so I assumed that the first 160GB would be my existing C drive and the second on would be the partition which I had labeled as linux (drive D). So I selected the linux partition and clicked on proceed/ install now, and then this error was shown - "No root file system defined".. and I cannot proceed after this.   Please let me know two things here:   During the screen in point number 3 above, the image with slider was shown. Which partition (drive c: or d: or anyother) is being referred to here? Which one am I actually resizing - is it the existing c: drive which has windows 7 installed already, that is getting shrunk and then ubuntu getting installed in the remaining space? What does ""No root file system defined" error mean? How do I select the proper partition and continue with the installation?    P.S.:  I had already partitioned my drives as mentioned above, and the D drive with 160GB was specifically left free for installing ubuntu. I would like to Select D drive during setup and install ubuntu here, while still keeping the already installed windows7 in C drive.  Thanks! What is rank deficiency, and how to deal with it? Fitting a logistic regression using  ends with   Error in mer_finalize(ans) : Downdated X'X is not positive definite.    A likely cause of this error is apparently rank deficiency. What is rank deficiency, and how should I address it? Is there a program to mount all of my drives automatically? Suppose that $f : \mathbb{C} \to \mathbb{C}$ is entire. Let $\Omega = \{ z : \frac{1}{2} &lt; |z| &lt; 2\}$. Furthermore, suppose that $ \mathcal{F} = \{f(kz) : k \in \mathbb{C}\}$ is a normal family of analytic functions defined on $\Omega$; the family $\mathcal{F}$ is normal on $\Omega$ iff each sequence from $\mathcal{F}$ has a subsequence converging uniformly on compact subsets of $\Omega$. Prove that $f$ is a polynomial.  I'm studying for my qualifying exam in complex analysis, and this question appears on an old exam. Usually, when I want to show that some entire $f$ is a polynomial, I just try to show that there exists some $N$ so that $f^{(k)}(0) = 0$, all $k \ge N$, but I'm having trouble getting that strategy to work here. I've been looking at the sequence $f(\frac{z}{n})$ trying to make something work, but have not had any success.   I would prefer to receive a hint or two, and then I'll try to post my own solution. Thanks in advance! What is potential energy truly? 101 positive integers placed on a circle A Pigeonhole Principle problem:     101 positive integers are placed on a circle whose sum is 300. Prove   that it is possible to choose from these numbers some consecutive   numbers whose sum is equal to 200.   (I don't know if the word 'consecutive' is appropriate in this case ,I mean that these numbers follow each other on that circle) Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$ A former lecturer of mine once explained why, from a syntactic point of view, the English rule that negation and questions are formed with the auxiliary do follows from other syntactic facts about English.  More precisely, if you gave a good syntactician not familiar with English a corpus of English sentences containing everything but questions and negated phrases, then s/he could infer grammatical rules from it (such as headed-ness of phrases, what kind of arguments what word takes, etc.) From some of these rules, then, s/he could see that something specific would be wrong with the sentences "Gave you the present to Mary?" and "He went not to the cinema."  Can someone explain what exactly this something is and how it follows from other sentences?  FYI, my lecturer did this in Government &amp; Binding with X-Bar theory. This may be more suitable a question for the proposed  but as it's not online yet I'm trying it here first.   Edit: the beta is online now at . Today I got what I presume was the update that should take care of Spectre and Meltdown. That didn't go very well. What do I do now? See screen photo.    I can't boot the 4.4.0-108-generic kernel because of a kernel panic. I'm running 16.04 on an Intel Core i5-4670K CPU @ 3.40GHz × 4. Euclidean Algorithm for GCD of polynomials I am struggling to use the Euclidean algorithm for polynomials.  Given something like $$GCD(x^5+1, x^3+1)$$  I can easily use it as follows:   $$x^5+1 = x^2(x^3+1) -x^2 +1 \\ x^3+1 = -x(-x^2+1) + x +1 \\ -x^2+1 = (x+1)(-x+1)$$  GCD = x+1  But for something like $$GCD(2x^2+6x+3, 2x+1)$$   I cannot figure out how to do it using the same method. I start like this:  $$2x^2+6x+3 = x(2x+1) + 5x+3\\ 2x+1 = \frac{2}{5}(5x+3) - \frac{1}{5}\\$$  and I am not sure how to continue. Any help would be appreciated, sorry for my poor attempt at using LaTeX. I only get four video output options Even when I start a brand new file in Blender, I only get four video output file options: AVI raw, AVI JPEG, Frame Server, FFMpeg Video.  I am running version 2.78.5 (I'm using the beta so that I can use denoising.  How can I get access to all the other video outputs other people have? Like this: