Finite group $(G, \cdot)$ and H a subgroup of G. Prove that if $x \notin H, y \notin H$ then $xy \in H$.
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
LibreOffice problem while openning a document My problem is that when I try to open one specific .odt document, I can't move the page or do anything except moving the mouse, which is useless. Also, the mouse, when I'm on the document, is freaking out switching between mouse and loader couple of times per second. It also becomes dark when I click on the document like if it's not responding. Why is the Java plugin (JRE) disabled in Chrome? Why is the Java plugin (JRE) is disabled in Chrome? It is some security concern?  From official Java website:     Chrome no longer supports NPAPI (technology required for Java applets)   The Java plug-in for web browsers relies on the cross platform plugin architecture NPAPI,   which has been supported by all major web browsers for over a decade.   Google's Chrome version 45 (scheduled for release in September 2015) drops support   for NPAPI,   impacting plugins for Silverlight, Java, Facebook Video   and other similar NPAPI based plugins.   But anyone knows why? How it could be dangerous for Chrome user with latest version of Java JRE installed? Collection was modified; enumeration operation may not execute I can't get to the bottom of this error, because when the debugger is attached, it does not seem to occur.  Collection was modified; enumeration operation may not execute  Below is the code. This is a WCF server in a Windows service. The method NotifySubscribers() is called by the service whenever there is a data event (at random intervals, but not very often - about 800 times per day). When a Windows Forms client subscribes, the subscriber ID is added to the subscribers dictionary, and when the client unsubscribes, it is deleted from the dictionary. The error happens when (or after) a client unsubscribes. It appears that the next time the NotifySubscribers() method is called, the foreach() loop fails with the error in the subject line. The method writes the error into the application log as shown in the code below. When a debugger is attached and a client unsubscribes, the code executes fine. Do you see a problem with this code? Do I need to make the dictionary thread-safe? [ServiceBehavior(InstanceContextMode=InstanceContextMode.Single)] public class SubscriptionServer : ISubscriptionServer {     private static IDictionary&lt;Guid, Subscriber&gt; subscribers;      public SubscriptionServer()     {                     subscribers = new Dictionary&lt;Guid, Subscriber&gt;();     }      public void NotifySubscribers(DataRecord sr)     {         foreach(Subscriber s in subscribers.Values)         {             try             {                 s.Callback.SignalData(sr);             }             catch (Exception e)             {                 DCS.WriteToApplicationLog(e.Message,                    System.Diagnostics.EventLogEntryType.Error);                  UnsubscribeEvent(s.ClientId);             }         }     }          public Guid SubscribeEvent(string clientDescription)     {         Subscriber subscriber = new Subscriber();         subscriber.Callback = OperationContext.Current.                 GetCallbackChannel&lt;IDCSCallback&gt;();          subscribers.Add(subscriber.ClientId, subscriber);                  return subscriber.ClientId;     }      public void UnsubscribeEvent(Guid clientId)     {         try         {             subscribers.Remove(clientId);         }         catch(Exception e)         {             System.Diagnostics.Debug.WriteLine(&quot;Unsubscribe Error &quot; +                      e.Message);         }     } } Let  $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.    I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as  $g(xy) = g(x) + g(y)$   I looked up the hint and it says let $g(x) = \log f(a^x) $   The wikipedia page for functional equations only states the form of the solutions without proof.  Attempt  Using the hint (which was like pulling a rabbit out of the hat)  Restricting the codomain $f:(0,+\infty)\rightarrow (0,+\infty)$ so that we can define the real function $g(x) =  \log f(a^x)$ and we have $$g(x+y) = g(x)+ g(y)$$   i.e $g(x) = xg(1)$ as $g(x)$ is continuous (assuming $f$ is).  Letting $\log_a f(a) = p$ we get $f(a^x) =a^p $. I do not have a rigorous argument but I think I can conclude that $f(x) = x^p$ (please fill any holes or unspecified assumptions) Different solutions are invited I am rebuilding an old server.  Only one of the five old sites need to go live and all five of the old sites point to the same IP address.  Unfortunately any time a request is made for one of the old sites via their own unique domain names they are passed the pages from the one site that I'm trying to make live.  I would just like the old sites to display a 404 instead of the one site that I'm actually trying to make live.  As an example:   Site 1 (192.168.0.1) [www.first.com] Site 2 (192.168.0.1) [www.second.com] Site 3 (192.168.0.1) [www.third.com]   I want to move site 1 live again.  So I turned on apache on the machine with the IP address of 192.168.0.1, with all of site 1's files in the /var/www directory.  When I browse to 192.168.0.1 or www.first.com I correctly see site 1. When I browse to www.second.com and www.third.com I also receive site 1 which is the problem.  Is there a way I can 404 any request which comes in without www.first.com?  Here is my sites-available:  NameVirtualHost *:80  LogFormat "%h %l %u %t \"%r\" %&gt;s %b" common  &lt;Files ~ "^\.ht"&gt;     Order allow,deny     Deny from all     Satisfy All &lt;/Files&gt;  &lt;Directory /&gt;   Order Deny,Allow   Deny from all   Options None   AllowOverride None   Options None &lt;/Directory&gt;  &lt;Directory /var/www&gt;   Order Allow,Deny   Allow from all &lt;/Directory&gt;  &lt;VirtualHost *:80&gt;         ServerName first.com         ServerAlias www.first.com         DocumentRoot /var/www         ErrorLog /var/log/first.com.err         CustomLog /var/log/first.com.log common &lt;/VirtualHost&gt;  &lt;VirtualHost _default_:*&gt;         ServerName unknown         DocumentRoot /var/www_404 &lt;/VirtualHost&gt; Surjectivity implies injectivity and conversely I have the low battery alarm disabled in the BIOS settings on my Lemur laptop, but the alarm still sounds when the battery is low.  The sound is a very loud and annoying beep that repeats every couple seconds. The laptop will last another 20 minutes without being plugged in so the battery level isn't properly being detected anymore anyways. I found that if I mute my audio then the beep stops but I obviously cannot listen to anything anymore. If i turn the volume back on even to just 1% the battery alarm is audible again at full volume.  I remember there used to be a mixer somewhere that could control the internal speaker and so if I could find that again then maybe i could mute that and fix this problem.   I have Ubuntu 14.04.  Note that I'm not talking about a low battery notification, which is something through gnome; as opposed to a low battery ALARM which is this very loud beep every couple seconds coming through the internal speaker.  Nevertheless, here are my gnome power settings:  $ gsettings list-recursively org.gnome.settings-daemon.plugins.power org.gnome.settings-daemon.plugins.power button-power 'interactive' org.gnome.settings-daemon.plugins.power critical-battery-action 'nothing' org.gnome.settings-daemon.plugins.power percentage-low 1 org.gnome.settings-daemon.plugins.power priority 0 org.gnome.settings-daemon.plugins.power lid-close-suspend-with-external-monitor false org.gnome.settings-daemon.plugins.power idle-dim false org.gnome.settings-daemon.plugins.power button-hibernate 'hibernate' org.gnome.settings-daemon.plugins.power sleep-inactive-ac-type 'suspend' org.gnome.settings-daemon.plugins.power button-sleep 'suspend' org.gnome.settings-daemon.plugins.power button-suspend 'suspend' org.gnome.settings-daemon.plugins.power sleep-inactive-battery-timeout 0 org.gnome.settings-daemon.plugins.power time-low 1200 org.gnome.settings-daemon.plugins.power lid-close-ac-action 'nothing' org.gnome.settings-daemon.plugins.power notify-perhaps-recall true org.gnome.settings-daemon.plugins.power percentage-critical 1 org.gnome.settings-daemon.plugins.power percentage-action 1 org.gnome.settings-daemon.plugins.power sleep-inactive-battery-type 'suspend' org.gnome.settings-daemon.plugins.power time-action 120 org.gnome.settings-daemon.plugins.power lid-close-battery-action 'nothing' org.gnome.settings-daemon.plugins.power idle-brightness 30 org.gnome.settings-daemon.plugins.power sleep-inactive-ac-timeout 0 org.gnome.settings-daemon.plugins.power time-critical 300 org.gnome.settings-daemon.plugins.power active true org.gnome.settings-daemon.plugins.power use-time-for-policy false $ I'm not sure how to prove these basic theorems in propositional calculus. Instead of using the standard axioms, we're supposed to use:   Deduction Theorem (if $\Phi, \alpha \vdash \beta$ then $\Phi \vdash \alpha \to \beta$),  Reductio (if $\Phi, \alpha \vdash \, $, then $\Phi \vdash \lnot \alpha$), Cut Rule (if $\Phi \vdash \alpha$ and $\Psi, \alpha \vdash \beta$ then $\Phi \cup \Psi \vdash \beta$),  Inconsistency Effect (if $\Phi \vdash \, $, then $\Phi \vdash \beta$ for every formula $\beta$), and  the Principle of Indirect Proof (if $\Phi, \lnot \alpha \vdash \, $, then $\Phi \vdash \alpha$),    as all the axioms can be deduced using these theorems.  I don't really know how to start the proofs without using the axioms:  i) prove that $\lnot(\alpha \to \beta) ⊢ \alpha$  ii) prove that $\lnot\alpha \vdash \alpha \to \beta$  Any suggestions on how to start these proofs or any insight at all would be greatly appreciated!  Thanks! Why doesn't the UK ask for travel bookings for a tourist visitor visa? Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$ How would one go about proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where  $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$?    I am attempting to use the fundamental theorem of finitely generated abelian groups but am struggling. In the interest of honesty, this is a past exam question that I am attempting for which the solutions are not available. EDIT: As the question was asked in the paper, the Chinese Remainder Theorem would not be permitted since this is only proved in the follow-up course. Firstly my main goal is to try and get FTP files recursively in Centos.  However the actual FTP login contains an @ which ruins the syntax.  Is there some way to place the username in "double quotes" or some such equivalent such that I can use @in the username? I've been reading up on STL containers in my book on C++, specifically the section on the STL and its containers. Now I do understand each and every one of them have their own specific properties, and I'm close to memorizing all of them... But what I do not yet grasp is in which scenario each of them is used.  What is the explanation? Example code is much prefered. Adding other css files to wp_head 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! Let's say I'm trying to answer a lot of questions just to up my reputation (even if I generally don't know what I'm talking about).  Then, whenever I get downvoted, I just delete my answer so that nobody sees that I've posted stupid answers, and so I don't lose reputation points.   Is this frowned upon? Do I lose reputation points on downvoted answers if I delete them? Accessing UI thread handler from a service Maximum power transfer and Thevenin I am trying to find the thevenin resistance value and the Vth for maximum power transfer. I have tried to find Rth by calculating R3 and R2 as parallel resistance and same for R4 and R5 and then adding them with R1 but I am not sure if I have done it the right way. For Vth I have tried using KVL but I think I have calculated wrong current values using current divider rule. I calculated the total current by diving V1 and R1. I hope someone can help. I have attached the picture of the circuit. Is there a term for words that when reversed, form other words? I'm aware of what a palindrome is. What do you call words that, when reversed, form other words, such as ton (not) and part (trap)? I'm using NetBeans, trying to change the familiar Java coffee cup icon to a png file that I have saved in a resources directory in the jar file.  I've found many different web pages that claim they have a solution, but so far none of them work.  Here's what I have at the moment (leaving out the try-catch block):  URL url = new URL("com/xyz/resources/camera.png"); Toolkit kit = Toolkit.getDefaultToolkit(); Image img = kit.createImage(url); getFrame().setIconImage(img);   The class that contains this code is in the com.xyz package, if that makes any difference.  That class also extends JFrame.  This code is throwing a MalformedUrlException on the first line.  Anyone have a solution that works? Using Rule of Three to obtain confidence interval for a binomial population I have a large population of data instances (say, 1000 instances) that are either of class1 or of class2. I would like to obtain a confidence interval for how many instances are of class1 without exhaustively checking all instances. I have randomly sampled 50 instances, and all 50 were of class1. I used the  to determine that a 95% confidence interval for the percentage that an instance is of class1 is  [0.94, 1].  From my sampling, I know that at least 50 instances are of class1. For the remaining 1000 – 50 = 950 instances whose classes are unknown, I assume I can apply the [0.94, 1] confidence interval found above. Therefore, can I conclude that, with a 95% confidence, there are at least 50 + (1000 – 50)(0.94) = 943 instances from the population of 1000 that are of class1?  If this conclusion isn’t statistically sound, how can I obtain a confidence interval for class1?