How to find $\left|\operatorname{Aut}(\mathbb Z_2\times\mathbb Z_2)\right|=?$  Since $\mathbb Z_2\times\mathbb Z_2$ has 4 elements, $1 \leq \left|\operatorname{Aut}(\mathbb Z_2\times\mathbb Z_2)\right|\leq \left|S_4\right|$.  How to proceed further?  Thanks in advance...
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...
In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$ Is there a way to do concurrent editing? Is there a MediaWiki-like method for supporting concurrent editing of text bodies in Drupal 7?  If no feature currently exists, what are some decent PHP diff libraries that could be wrapped up by a Drupal module that could do the job? I often read outputs in the terminal that get wrapped as they are too long. Since there is no way that I can modify the printed output, I need to let the Terminal allow for horizontal scrolling.  Is there a modification I can make to the default OS X Terminal to disable line wrapping and allow for horizontal scrolling?  Or do you know another alternative I can use? Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable? Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)&gt;c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable?  There are a number of equivalent definitions for the measurability of a function and the most obvious one would be to show that $\{x\in E : f(x)&gt;c\}$ is measurable for all $c\in\mathbb{R}$.  Thus my strategy has been to consider an arbitrary irrational $y$ and use the density of the rationals in the reals to show that there exists some open set $O$ such that $m^*(O-\{x\in E : f(x)&gt;y\})&lt; \varepsilon$.  I would do this by choosing some rational $q&lt;y$, close enough to $y$, such that the set of values in $E$ which get sent under $f$ to values strictly greater than $q$ and less than or equal to $y$ is small enough.  But that doesn't in general seem possible since I could keep sending intervals of some fixed positive length to smaller and smaller regions just beneath $y$.  However conversely, such a function would not be enough for a counter example, since finding such a function only rules out this strategy, but doesn't necessarily entail non-measurability.  Is there a better definition of the measurability of a set I should use? or does the statement actually not imply measurability?  Thanks. Unmet dependencies unable to fix I'm running ubuntu 14.04. I cannot use software center to install anything, even a local deb package so I have to rely on Terminal.  Recently I was trying to install mysql-server which generated error which I could pin-point was the required version of libstdc++6 . My system had 4.8.x while mysql 5.6 required >=4.9. Trying the usual upgrading process and unsuccessful, I consulted my sys admin, who tried installing the 4.9 package manually(using a local .deb package), which failed as well citing unmet dependencies.   At present, what I have is a system with error saying no new software can be installed.  The sudo apt-get install -f gives following output:  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... failed. The following packages have unmet dependencies:  libstdc++6 : Depends: gcc-4.9-base (= 4.9.2-10) but 4.9.1-0ubuntu1 is installed  libstdc++6:i386 : Depends: gcc-4.9-base:i386 (= 4.9.2-10) but 4.9.1-0ubuntu1 is installed E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.   I checked for held packages using  dpkg -l | grep ^h dpkg -l | grep hold   Result: nothing.  Also tried to find the broken package, if any but the file  /var/log/dist-upgrade/apt.log doesn't exist.  So, what should/can I do now. Is there a difference between single and double quotes in Java? Is there a difference between single and double quotes in Java? Contradicting Fubini's theorem I have a function defined as follows: $f(x,y)= \dfrac{x^2-y^2}{\left(x^2+y^2\right)^2}$, if $(x,y)\neq (0, 0)$ and $f(x,y)=0$ if $(x,y)=(0,0)$.  Now, $$\int_0^1\int_0^1 f(x,y)\,\text{d}x~\text{d}y=-\frac{\pi}{4}$$ and $$\int_0^1\int_0^1 f(x,y)\,\text{d}y~\text{d}x=\frac{\pi}{4}.$$   The question I have is this: why does this not contradict Fubini's theorem?    thanks. Is it possible to skip Skyrim's new game introduction cinematic? I don't want to watch the prisoner cart ride to the execution: is it possible to skip this?   If so, how? Can I replace the intro movie file by some empty video? basic geometry angle problem Two straight lines, AB and AD, lie in the same vertical plane. AB makes an angle of 25 degrees with the horizontal, and AD makes 32 degrees with the vertical. What is the angle between AB and AD?  The answer is 33 degrees or 93 degrees.   I cannot visualize what this question is asking. It says both AD and AD lie on the same vertical plane, then immediately says AD makes 32 degrees with the vertical. Isn't this a contradiction? Block any query in BIND I've been trying to block ANY queries in my BIND server with no success.  I followed this post  which says that I can use minimal-responses but it doesn't work.  None of the answers fit my current configuration.  view world {     match-clients   { world-clients; };     allow-recursion { none; };     recursion       no;     allow-transfer  { key XXXXXX; };      minimal-responses yes;      forwarders      { };      include         "/etc/bind/world.conf"; }; Currently using Ubuntu 18.04, and my wireless driver is not being detected.  I've included a .  $ dmesg | grep iwl [    6.212724] Loading modules backported from iwlwifi [    6.212724] iwlwifi-stack-public:master:6988:0d8892f2 [    6.224583] iwlwifi 0000:00:14.3: enabling device (0000 -&gt; 0002) [    6.229811] iwlwifi 0000:00:14.3: Direct firmware load for iwl-dbg-cfg.ini failed with error -2 [    6.230540] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-39.ucode failed with error -2 [    6.230547] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-38.ucode failed with error -2 [    6.230552] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-37.ucode failed with error -2 [    6.230557] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-36.ucode failed with error -2 [    6.230562] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-35.ucode failed with error -2 [    6.236670] iwlwifi 0000:00:14.3: loaded firmware version 34.0.0 op_mode iwlmvm [    6.248314] iwlwifi 0000:00:14.3: Detected Intel(R) Dual Band Wireless AC 9560, REV=0x318 [    7.264081] iwlwifi 0000:00:14.3: SecBoot CPU1 Status: 0x3, CPU2 Status: 0x2477 [    7.264084] iwlwifi 0000:00:14.3: Failed to start INIT ucode: -110 [    7.276165] iwlwifi 0000:00:14.3: Failed to run INIT ucode: -110   sudo modprobe iwlwifi returns no output. Proof that every number ≥ $8$ can be represented by a sum of fives and threes. Can you check if my proof is right?  Theorem.  $\forall x\geq8, x$ can be represented by $5a + 3b$ where  $a,b \in \mathbb{N}$.  Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\ x=9 = 3\cdot3 + 5\cdot0 \quad \checkmark\\ x=10 = 3\cdot0 + 5\cdot2 \quad \checkmark$  Inductive step:  $n \in \mathbb{N}\\a_1 = 8, a_n = a_1 + (x-1)\cdot3\\ b_1 = 9, b_n = b_1 + (x-1)\cdot3 = a_1 +1 + (x-1) \cdot 3\\ c_1 = 10, c_n = c_1 + (x-1)\cdot3 = b_1 + 1 + (x-1) \cdot 3 = a_1 + 2 + (x-1) \cdot 3\\ \\ S = \{x\in\mathbb{N}: x \in a_{x} \lor x \in b_{x} \lor x \in c_{x}\}$  Basis stays true, because $8,9,10 \in S$  Lets assume that $x \in S$. That means $x \in a_{n} \lor x \in b_{n} \lor x \in c_{n}$.  If $x \in a_n$ then $x+1 \in b_x$,  If $x \in b_x$ then $x+1 \in c_x$,  If $x \in c_x$ then $x+1 \in a_x$.  I can't prove that but it's obvious. What do you think about this? Why is __init__() always called after __new__()? I'm just trying to streamline one of my classes and have introduced some functionality in the same style as the .  However, I'm a bit confused as to why __init__ is always called after __new__. I wasn't expecting this. Can anyone tell me why this is happening and how I can implement this functionality otherwise? (Apart from putting the implementation into the __new__ which feels quite hacky.)  Here's an example:  class A(object):     _dict = dict()      def __new__(cls):         if 'key' in A._dict:             print "EXISTS"             return A._dict['key']         else:             print "NEW"             return super(A, cls).__new__(cls)      def __init__(self):         print "INIT"         A._dict['key'] = self         print ""  a1 = A() a2 = A() a3 = A()   Outputs:  NEW INIT  EXISTS INIT  EXISTS INIT   Why? After I deleted my Ubuntu partition, I rebooted my computer and I only get a prompt:   grub rescue &gt;   The command ls lists   (hd0) (hd0,msdos5) (hd0,msdos3) (hd0,msdos2) (hd0,msdos1) (hd1) (hd1,msdos1)   I check them one by one:  ls (hd0,msdos5)/ ls (hd0,msdos1)/   But I cannot find the right partition. Please help identify this 2-part TV movie: Alien woman crash-lands on earth, falls in love, 199x How are connections differentiated from behind NAT routers? Which Distribution Does the Data Point Belong to? Let me start by explaining my particle physics background is very patchy, so this question may not be as coherent as  I would like it to be.   In general terms, what is the difference between a resonance found during a particle experiment and a particle?   From reading Wikipedia it seems to be mostly based on timescales. If a heavy particle decays quickly, how can we distinguish it from a resonance?  I would not be posting this question except for the fact that the Wikipedia page says "This page has some issues".        and I would be interested in knowing more about the subject. Proving the two given vectors are orthogonal I am given the vectors $\mathbf{w}, \mathbf{v}, \mathbf{u}$ in $\mathbb{R}^n$ such that $\mathbf{u} \neq 0$ and $\mathbf{w} = \mathbf{v} - \frac{\mathbf{u}\,\bullet\,\mathbf{v}}{\|\mathbf{u}\|^2}\bullet\mathbf{u}$. I am asked to show that the vector $\mathbf{w}$ is orthogonal to $\mathbf{u}$.     So far, I have written out the definition of orthogonal: two vectors are orthogonal if and only if their dot product is zero.   So what we need to prove is $\mathbf{w}\bullet\mathbf{u} = 0$ where $\mathbf{w}\bullet\mathbf{u}$ is defined as $\mathbf{w}^T\bullet\mathbf{u}$.   However, I have been stuck on this problem for about an hour and haven't made any significant progress from here. How do we go about proving that the vectors are orthogonal? Find number of files with a specific extension, in all subdirectories Is there a way to find the number of files of a specific type without having to loop through all results inn a Directory.GetFiles() or similar method? I am looking for something like this:  int ComponentCount = MagicFindFileCount(@"c:\windows\system32", "*.dll");   I know that I can make a recursive function to call Directory.GetFiles , but it would be much cleaner if I could do this without all the iterating.  EDIT: If it is not possible to do this without recursing and iterating yourself, what would be the best way to do it?