Prove that for any prime $p$ there is no rational number $a$ such that $a^2 = p$.
use contradiction to prove that the square root of $p$ is irrational
Why doesn't hydrogen gas exist in Earth's atmosphere? I see questions/information online about how to get rid of corpses but I'm not sure what the consequences to having them lying around are.  I know that I can use a morgue to store them and that they will eventually be taken away in a hearse. However, I've always just left them lying around. This doesn't seem to have any obvious consequences.  Do they upset prisoners or staff (e.g. increasing the danger level beyond what death normally does)? Make them sick? Slow down pathing? Is this something that's not implemented yet or is leaving dead bodies all over the place meant to be completely fine?  The wiki's pages for  and  don't shed any light on this. All I'm finding is the how of getting rid of corpses/glitches related to it but not why it matters in the first place. All I've seen that's related is that apparently prisoners can loot guard corpses for keys, but there's no obvious consequences for prisoner corpses which are what I tend to end up with.  So why not just leave them in the corridors? How to differentiate formally? What does 1x1 convolution mean in a neural network? I am currently doing the Udacity Deep Learning Tutorial. In Lesson 3, they talk about a 1x1 convolution. This 1x1 convolution is used in Google Inception Module. I'm having trouble understanding what is a 1x1 convolution.  I have also seen  by Yann Lecun.  Could someone kindly explain this to me? Is there a way to add some custom font on a website without using images,  or some other graphics?  For example, I was working on a wedding website, and I found a lot of nice fonts for that subject. But I can't find the right way to add that font on the server. And how do I include that font with CSS into the HTML? Is this possible to do without graphics? $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ . $\operatorname{Aut}(V)$ is isomorphic to $S_3$ I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them ).  The lectures come complete with notes and homework problems.  Of course, since I don't actually go to Harvard, I can't hand in the homework assignments to find out if I'm doing them correctly.  So I've decided to try to crowd source the grading of my solutions.  I'll post individual questions as I finish them and wait for comments.  I would like to get critiques of not just my reasoning but the style of my write ups as well.  Also, any alternative approaches to the problem would be welcome.  I've looked for forums dedicated to these kinds of OCW courses but have not been able to find any.  This surprises me.  It seems like, with the advent of these free online educational resources, an online meeting place for those who take advantage of them would be a natural offshoot.  So this might be a bit of an experiment.  Unless I'm the 873rd person to post something like this here.  If that's the case, sorry for being so long winded.  Anyway, on with the question.  This one is assigned in the 4th lecture.     Let $V$ denote the Klein 4-group.  Show that $\operatorname{Aut}(V)$ is isomorphic to $S_3$.   (I've moved my solution below to keep this question from showing up in the unanswered list.)  Thanks... Can't delete folder in Windows 7 Why does an object with a glossy shader render in black? When I apply a glossy shader to the material of my text, the text becomes just black. Why?    I have lamp and everything, but the problem persists. The text is red before I add the glossy shader.  I'm using Cycles render.   Thanks! Distance between surfaces Probability associated with experiencing all outcomes Can an Indian travel to India with 50 days before passport expires? My wife is travelling from Philadelphia to India in January 2019 and, on her outbound trip, she will only have 50 days validity left on her passport. However, while in India, she will be renewing her passport. The reply from a similar question seems to answer point 1 below, but I wanted a second opinion to be sure.  Here are the questions we have:  Can she travel from the US to India via Doha, Qatar with passport validity of only 50 days? Can we make a reservation on Qatar Airways for the roundtrip travel using her current passport and before the return from India to the US update passport information on the reservation? How do I install pillow in Ubuntu 16.04 LTS? I went to  for installing Pillow since the original PIL is a discontinued project and so now not so good to use. But, in the table support for Ubuntu 16.04 LTS is unmentioned. It appears neither in , or in .     Further, just installing pillow is said to be not enough. There are some dependencies that should be installed first which is also unmentioned for Ubuntu 16.04 LTS. Would the installation steps for Pillow for python 3 given for 14.04 LTS be OK for Ubuntu 16.04 LTS?  If not, How do I install pillow in 16.04 LTS for Python 3.6? (There is support for Python 2.) Scale Insetted Characters to Plot Is there any point in taking the alternate exits in Bit Trip Runner 2?  I can't see any benefit to doing so in terms of unlocking a different path in the overworld, so is it just for your own benefit? What is the CVE-2014-6271 bash vulnerability (Shellshock) and how do I fix it? Recently, there have been news going around regarding "CVE-2014-6271" (See ), which is a vulnerability in Bash. How do I know if I am affected by this, how can I fix it, and why should I care?  This is designed as a canonical answer for this vulnerability, due to its scope and severity. What is the connection between regularization and the method of lagrange multipliers ? To prevent overfitting people people add a regularization term (proportional to the squared sum of the parameters of the model) with a regularization parameter $\lambda$ to the cost function of linear regression. Is this parameter $\lambda$ the same as a lagrange multiplier? So is regularization the same as the method of lagrange multiplier? Or how are these methods connected? Atheros AR9462 WiFI very unstable & package loss I'm having severe issues w/ package-loss and a generally bad connection w/in a WiFi-network at my uni. I'm usually encountering a package loss of ~50% and aborting connections which makes working w/ SaaS impossible.  I'm using Ubuntu 13.04 and my network-adapter is an Atheros AR9462.  Applying the nohwcrypt=1-option doesn't change anything.  Also very noticable: The connection works as intended in other networks; the problematic network uses WPA2 w/ MSCHAPv2.  Anyone can help my out here?  $ sudo lshw -class network  *-network UNCLAIMED             description: Ethernet controller        product: NetLink BCM57780 Gigabit Ethernet PCIe        vendor: Broadcom Corporation        physical id: 0        bus info: pci@0000:04:00.0        version: 01        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list        configuration: latency=0        resources: memory:d3500000-d350ffff   *-network        description: Wireless interface        product: AR9462 Wireless Network Adapter        vendor: Atheros Communications Inc.        physical id: 0        bus info: pci@0000:09:00.0        logical name: wlan0        version: 01        serial: e0:06:e6:76:ba:89        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list rom ethernet physical wireless        configuration: broadcast=yes driver=ath9k driverversion=3.8.0-22-generic firmware=N/A ip=131.173.74.130 latency=0 link=no multicast=yes wireless=IEEE 802.11abgn        resources: irq:19 memory:d3400000-d347ffff memory:9fb00000-9fb0ffff 8 bits representing the number 7 look like this:  00000111   Three bits are set.     What are algorithms to determine the number of set bits in a 32-bit integer? ASP.NET MVC 3 localization with DisplayAttribute and custom resource provider