There is one rocket in each corner of a square. At some point they start moving towards the rocket in the neighbouring corner in (say) clockwise direction. Their subsequent motion is such that they move exactly in the direction of the rocket they "follow" at any moment. The magnitude of their speed is constant (and is the same for any rocket). They all collide at the center of the square (as can be seen intuitively). What is the length of their path taken before the collision?  I heard that it has a very easy and a complicated solution too (with integrals). Could you give both and tell if it can be generalized for regular n-gons?  The answer is the length of the sides of the square but I don't know how we arrive at it.
This is from a problem seminar and I need help figuring out the solution.  Four bugs, $A,B,C,D$ are initially placed at the corners of a unit square. From a given initial moment, all four crawl simultaneously at one and the same speed $s&gt;0$, $A$ towards $B$, $B$ towards $C$, $C$ towards $D$, and $D$ towards $A$, with each heading at every instant along the line joining it with its target.  Find a system of differential equations describing the trajectory of $A$.  Do the bugs eventually all meet at the center of the square? If so, how long did it take?  My attempt so far: $\def\grad{\mathbf\nabla}$ $\newcommand{\norm}[1]{\left\Vert #1 \right\Vert}$ Call ${\vec x}_1,{\vec x}_2,{\vec x}_3,{\vec x}_4$  the positions in the plane of $A,B,C,D$ respectively. Then $$\grad {\vec x}_1 = \frac{{\vec x}_2-{\vec x}_1}{\norm{{\vec x}_2-{\vec x}_1}}s$$ $$\grad {\vec x}_2 = \frac{{\vec x}_3-{\vec x}_2}{\norm{{\vec x}_3-{\vec x}_2}}s$$  $$\grad {\vec x}_3 = \frac{{\vec x}_4-{\vec x}_3}{\norm{{\vec x}_4-{\vec x}_3}}s$$  $$\grad {\vec x}_4 = \frac{{\vec x}_1-{\vec x}_4}{\norm{{\vec x}_1-{\vec x}_4}}s$$  with ${\vec x}_1(0) = (0,0),\  {\vec x}_2(0) = (0,1),\ {\vec x}_3(0) = (1,1),\ {\vec x}_4(0) = (1,0)$.  I have noticed that the sum of the gradients is 0, but I don't know if that is helpful or not. Can someone help me figure this problem out?
I have an alias that I would like to use both as a regular user and as root, via sudo. Specifically, it is this:  alias rm=trash   This works fine as a normal user, and it works fine when I use sudo -i to get a root shell prompt, but if I use sudo rm, the alias does not apply. So where do I need to put my alias so that it works in one-off sudo commands? For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution. For a prime $p \equiv 1$ or $3$  (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.   I'd like to see an elementary proof. Why I cannot ping 192.168.0.108 (my computer IP?) Ipconfig says that my IP is 192.168.0.108 C:\Users\vbcbu&gt;Reply from 192.168.0.108: Destination host unreachable.ip 'Reply' is not recognized as an internal or external command, operable program or batch file.  C:\Users\vbcbu&gt;ipconfig  Windows IP Configuration   Ethernet adapter Ethernet:     Media State . . . . . . . . . . . : Media disconnected    Connection-specific DNS Suffix  . :  Wireless LAN adapter Local Area Connection* 12:     Media State . . . . . . . . . . . : Media disconnected    Connection-specific DNS Suffix  . :  Wireless LAN adapter Local Area Connection* 13:     Media State . . . . . . . . . . . : Media disconnected    Connection-specific DNS Suffix  . :  Wireless LAN adapter Wi-Fi 2:     Connection-specific DNS Suffix  . : www.tendawifi.com    Link-local IPv6 Address . . . . . : fe80::3804:2d04:a591:7142%6    IPv4 Address. . . . . . . . . . . : 192.168.0.108    Subnet Mask . . . . . . . . . . . : 255.255.255.0    Default Gateway . . . . . . . . . : 192.168.0.1  Ethernet adapter Bluetooth Network Connection:     Media State . . . . . . . . . . . : Media disconnected    Connection-specific DNS Suffix  . :  C:\Users\vbcbu&gt;  I can ping router at 192.168.0.1 I can't ping 192.168.0.108 C:\Users\vbcbu&gt;ping 192.168.0.1  Pinging 192.168.0.1 with 32 bytes of data: Reply from 192.168.0.1: bytes=32 time=2ms TTL=64 Reply from 192.168.0.1: bytes=32 time=24ms TTL=64  Ping statistics for 192.168.0.1:     Packets: Sent = 2, Received = 2, Lost = 0 (0% loss), Approximate round trip times in milli-seconds:     Minimum = 2ms, Maximum = 24ms, Average = 13ms Control-C ^C C:\Users\vbcbu&gt;ping 192.168.0.8  Pinging 192.168.0.8 with 32 bytes of data: Reply from 192.168.0.108: Destination host unreachable. Reply from 192.168.0.108: Destination host unreachable. Reply from 192.168.0.108: Destination host unreachable. Reply from 192.168.0.108: Destination host unreachable.  Ping statistics for 192.168.0.8:     Packets: Sent = 4, Received = 4, Lost = 0 (0% loss),  C:\Users\vbcbu&gt;ping 192.168.0.8  Pinging 192.168.0.8 with 32 bytes of data: Reply from 192.168.0.108: Destination host unreachable. Reply from 192.168.0.108: Destination host unreachable. In chapter 2 section 7 (pg 151) of Hartshorne's algebraic geometry there is an example given that talks about automorphisms of $\mathbb{P}_k^n$. In that example Hartshorne states that $\mathcal{O}(-1)$ has no global sections. However, we know that $\mathcal{O}(1)$ is generated by global sections. This is stated at the first of the section that if $\mathbb{P}_k^n = Proj k[x_0,...,x_n]$ then the $x_0,...,x_n$ give rise to global sections $x_0,...,x_n\in\Gamma(\mathbb{P}_k^n,\mathcal{O}(1))$.  I guess I don't understand this twisted structure sheaf very well, or to be honest I don't think I understand structure sheaves in general as well as I would like. The twisting part seems simple at first- you shift the grading of Proj over and then take the structure sheaf.  Admittedly I don't feel comfortable using the structure sheaf other than using the basic facts about it that Hartshorne gives when it is introduced. If anyone could give some insight as to what's going on here or how I might be able to understand this better it would be much appreciated. Thanks. $C^1[a,b]$ is closed in $C[a,b]$ My Linux Mint 15 install on VirtualBox was working great up until about two days ago, when the updater started throwing a ton of errors and refuses to progress with any apt-get update, apt-get install, or apt-get upgrade.  They are 404 errors on the archive.ubuntu.com site, under the /ubuntu/dists/raring-updates folder. I looked at it, it doesn't exist.  a direct link, which shows there is no raring-updates.  Where has raring-updates gone and what can I do to fix this? How does the subject-verb agreement work when a singular noun describes a plural one? Of the following, which sentence is righter?  The computers is a good choice. The computers are a good choice.  Is there a rule about these types of formations? Please explain. I found native people prefer to use passive voice in a sentence. For example:  Tonight's moon can be seen from anywhere worldwide. We have reason to believe that progress can be made.  Be honest, it's difficult for an English learner to understand. Why not phrase it this way:  We can see the moon from anywhere worldwide.  In which cases do people prefer to use passive voice rather than active voice? How do you prove the  from statistical mechanics? To prove entropy will only increase with time? How to prove? Please guide. Now let set $A$ be arbitrary. So set $A$ exists. Using axiom of pairing $\{A\}$ exists. And then using axiom of regularity, we can argue that $A \notin A$. So, there is some $x$ such that $x \notin A$. Is this a valid proof ? Thanks Book series about time travel from future to distant past, psychic powers? \documentclass[11pt]{article}  \begin{document}   The above equations on application of Laplace Transform become:  $$sL_1\hat{i}_{L1}(s) = D\cdot \hat{v}_{C1}(s) + D\cdot \hat{v}_{C2}(s) - D\prime \cdot \hat{v}_g(s) + (C_{C1} + V_{C2} - V_g + V_D)\cdot \hat{d}(s)$$ $$sL_2\hat{i}_{L2}(s) = -D\prime \cdot \hat{v}_{C1}(s) + D\cdot \hat{v}+{C2}(s) + D\prime \cdot \hat{v}_g(s) + (C_{C1} + V_{C2} - V_g + V_D)\cdot \hat{d}(s)$$ $$sC_1\hat{v}_{C1}(s) = -D\cdot \hat{i}_{L1}(s)+ D\prime \cdot \hat{i}_{L2}(s) + (-I_{L1} -I_{L2} - I_1)\cdot \hat{d}(s) -D\prime \cdot \hat{i}_l(s)$$ $$sC_1\hat{v}_{C1}(s) = D\prime \cdot \hat{i}_{L1}(s)- D\cdot \hat{i}_{L2}(s) + (-I_{L1} -I_{L2} - I_1)\cdot \hat{d}(s) -D\prime \cdot \hat{i}_l(s)$$ $$sL_l\hat{i}_l(s) = D\prime \cdot \hat{c}_{C1}(s) + D\prime \cot \hat{v}_{C2}(s) - D\prime \cdot \hat{v}_g(s) +(V_g - V_D - V_{C1} - V_{C2})\cdot \hat{d}(s) - R_l \cdot \hat{i}_l(s)$$  \end{document}   Error:   181 Underfull \hbox (badness 10000) in paragraph at lines 181--181 181 Overfull \hbox (22.63293pt too wide) detected at line 181 What is a "slug" in Django? $x,y,z$ are independent and uniform random on $[0,1]$.  I know that $P(x \geq yz) = 3/4$ by triple integrating the triple density function.  However, is there a easier way to get this solution? Without triple integration. Importing google-play-services lib into Intellij IDEA 12 (and 13) Merging multiple vector layers to one layer using QGIS Basic question about Merging faces I'm a beginner with blender so this is probably a very basic question.  I have this basic model below. I've ended up with a lot of unwanted geometry that I want to simplify. I wanted to start by merging the two selected faces below but no matter what I try to delete or merge it just destroys the geometry.  How can I achieve what I want to do (which is combining all of the unwanted geometry, either starting with this merge or some other way?) If so, what? If not, why not?  For a sample on the line, the median minimizes the total absolute deviation. It would seem natural to extend the definition to R2, etc., but I've never seen it. But then, I've been out in left field for a long time. Proper Installation of Kali linux tools in Ubuntu I think everyone has known - we need to install katoolin to use Kali Linux Tools on Ubuntu. Before did anything I had looked through the internet and also found out these solutions: ,   So here are my steps:  apt-get install git git clone https://github.com/LionSec/katoolin.git &amp;&amp; cp katoolin/katoolin.py /usr/bin/katoolin chmod +x  /usr/bin/katoolin sudo katoolin   Then, to add Kali Linux repositories and update them, I selected option 1 from the katoolin Menu  Then, to view the available categories and install all available tools, I selected option 2 from the main menu of katoolin and then pressed 0.   But after that I got this:    What do you want to do ?>    1) Add Kali repositories & Update  2) View Categories 3) Install classicmenu indicator 4) Install Kali menu 5) Help   kat > 2  ************************** All Categories **************************  1) Information Gathering                      8) Exploitation Tools 2) Vulnerability Analysis                     9) Forensics Tools 3) Wireless Attacks                           10) Stress Testing 4) Web Applications                           11) Password Attacks 5) Sniffing & Spoofing                        12) Reverse Engineering 6) Maintaining Access                         13) Hardware Hacking 7) Reporting Tools                            14) Extra  0) All   Select a category or press (0) to install all Kali linux tools .  Furthermore, I got an error in Software Updater - it didn't let me do anything. Ubuntu just hung up. I solved this problem by deleting kali linux repositories in katoolin and then in Software Updater.  Does anyone know how to install Kali Linux tools in Ubuntu 16.04 in a proper way? If cooking meat kills bacteria, and bacteria are responsible for problems with eating meat which has been left out, then why is it dangerous to eat meat which has been left out at room temperature and then thoroughly cooked?   mentions that "Even if the bacteria is dead, toxins can remain if the food was out too long, causing problems".  However, there's no further detail given.  Are these toxins as dangerous as the bacteria themselves?  How long does meat have to be left out to accumulate a dangerous level of toxins and thus be dangerous even if thoroughly cooked?  Are these toxins the reason for the usual guideline of keeping meat unrefrigerated for a maximum of 2 hours?  update  The revelation, courtesy of Aaronut, that e. coli is actually dangerous because of its toxins — which cannot be denatured at temperatures which will leave meat in an edible state — has pretty much answered this question.  And also given me further incentive to stop eating meat altogether :)  Our discussion (see the comments on hobodave's answer) has progressed into the realm of microbiology.  Some highlights from my ongoing research:  .  These seem to be the reason for the importance of keeping meat at a high temperature for a period of time.  .  This also provides fascinating insight into how bacteria evolved immunity to antibiotics.  Fungi.