prove by inductive step
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
Realtek RTL8723BE Wi-Fi incredibly weak I have a HP 14-AN001NA laptop. It came with Windows 10 installed on which the Wi-Fi worked fine anywhere in the house.   However I removed Windows 10 and installed Ubuntu 16.04 instead and since I have had terrible Wi-Fi anywhere aside from about three foot away from the router. I don't know if it's relevant but my Wi-Fi is not called wlan0 but wlo1 for some reason (I didn't change it).  When running lspci -knn | grep Net -A3     I get the following output:  05:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8723BE PCIe Wireless Network Adapter [10ec:b723] DeviceName:   Subsystem: Hewlett-Packard Company RTL8723BE PCIe Wireless Network Adapter [103c:81c1] Kernel driver in use: rtl8723be Kernel modules: wl, rtl8723be Clock time is off on dual boot Dual boot system Windows XP Pro and Ubuntu 12.04.  I have the bios set for the correct time and Ubuntu set for US Eastern time.  Ubuntu will boot up and the time will be off by -4 hours.  If I correct the time in Ubuntu then when I boot up in Windows XP the time will be off by + 4 hours.  This is a fresh install of 12.04. I did not have this problem before reinstalling Ubuntu. Short-cut for switching to a given directory Compiz Problems (Plugin Missing, Performance Issues, Development) I've recently set up the compiz cube again, and I was very happy to find that that flickering window glitch is gone! However, the cube reflection and deformation plugins are also gone.  I am running a fresh 12.10 install, and I have every compiz plugin package installed (including compiz-plugins-extra, which supposedly contains everything that compiz-fusion-plugins-extra package used to have), but those two plugins are still missing. I haven't been able to find anyone with the same problem.   Any ideas? (thanks!) I was looking at  and I didn't realize that Minecraft had some Easter eggs in it. Besides the Christmas chest Easter egg, and the sheep Easter egg, what else is there? My sister, cousins, and other relatives speculate that my mom's grandmother was Jewish (for various reasons). I've heard people say at times things like, for example, "I'm 50% German" or "I'm 25% Swedish." This has to do with their ancestry. But, how exactly is this calculated? Say for example that my mother's paternal grandmother was indeed 100% Jewish, what would that make me and my siblings, and how is it calculated? Show alert popup on standard page layout on click of standard button "Save" Chain length. how long or how many links I measured from the middle of each sprocket and it's 17 inches. Not sure what length of chain I should get How to change Function (Fn) Key behavior on wireless Logitech keyboard I do have a wireless Logitech K750 keyboard (silver Mac version) connected to a Lenovo ThinkPad (via the unifying USB receiver, together with a wireless Logitech mouse M705) and currently running Ubuntu 12.04.  By default the F-keys are mapped as multimedia/function keys, e.g. Play  /Pause. As I don't use them very often, I want the F-keys to behave like the "real" F-keys (F1-F12). How can I do that?  I know there is an option in the BIOS settings, but this only toggles the built-in keyboard of the notebook, not on my wireless keyboard.  I know there is a way to change this for the Apple wireless keyboard (), but there is no "parameters" directory in /sys/module/hid_logitech_dj/.  Any ideas? Or is there maybe a way to do this in a later Ubuntu version (13.04)?  UPDATE:  lsusb -d 046d: shows Bus 002 Device 005: ID 046d:c52b Logitech, Inc. Unifying Receiver  UPDATE 2: Thanks a lot for your help Lekensteyn. Finally  works just fine. In the "Known Issues" of Solaar it says that it does not work with the Nano Receiver. But I have one and it works just fine. Just don't forget to read the installation instructions and run the rules.d/install.sh script. It also recognized my M705 Mouse. Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. I define a map $$\phi:D_{2n} \rightarrow \mathbb{Z}_2 \times D_n$$ by $R \mapsto (0,r^{\frac{n+1}{2}})$ and $M \mapsto(1,m)$. Then I am stuck at showing the map is bijective. By the way, do we need to show that $(0,r^{\frac{n+1}{2}})(1,m)(0,r^{\frac{n+1}{2}})^{-1}=(1,m)$ ? If we can prove this, then what can we conclude? How do I lock a file in Perl? What is the best way to create a lock on a file in Perl?  Is it best to flock on the file or to create a lock file to place a lock on and check for a lock on the lock file? Dimensionality of null space when Trace is Zero This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216).  I've succeeded in proving the first three parts, but the most interesting part of the problem eludes me.  Part (a) requires the reader to prove that $\operatorname{Tr}{(AB)} = \operatorname{Tr}{(BA)}$, which I was able to show by writing out each side of the equation using sigma notation.  Part (b) asks the reader to use part (a) to show that similar matrices have the same trace.  If $A$ and $B$ are similar, then  $\operatorname{Tr}{(A)} = \operatorname{Tr}{(S^{-1}BS)}$  $= \operatorname{Tr}(BSS^{-1})$  $= \operatorname{Tr}(B)$,  which completes part (b).  Part (c) asks the reader to show that the vector subspace of matrices with trace equal to zero have dimension $n^2 - 1$.  Curtis provides the hint that the map from $M_n(F)$ to $F$ is a linear transformation.  From this, I used the theorem that $\dim T(V) + \dim n(T) = \dim V$ to obtain the dimension of the null space.  Part (d), however, I'm stuck on.  It asks the reader to show that subspace described in part (c) is generated by matrices of the form $AB - BA$, where $A$ and $B$ are arbitrary $n \times n$ matrices.  I tried to form a basis for the subspace, but wasn't really sure what it would look like since an $n \times n$ matrix has $n^2$ entries in it, but the basis would need $n^2 - 1$ matrixes.  I also tried to think of a linear transformation whose image would have the form of $AB - BA$, but this also didn't help me.  I'm kind of stuck...  Many thanks in advance! Help with Dual Javascript countdown clock I am having a little trouble figuring out how to make this work properly, I am trying to put a small countdown clock in the left column and one in the right column, which will show a countdown to two separate dates. I was able to create a single countdown clock using the below code and was hoping to be able to be able to tweak the code enough to allow it to run on a second CEWP with a different date. Is it possible to run two Javascript files on the page without them interferring with eachother? Thank you anyone that can lend a quick hand.      &lt;style type="text/css"&gt; .container{     text-align: center;     background: #f63536;   font-family: sans-serif;   font-weight: 100;   height:115px;   width:215px; }  h3{   color: #FFFFFF;   font-weight: 100;   font-size: 16px;   margin: 0px 0px 5px;   padding-top:3px; }  #clockdiv{     font-family: sans-serif;     color: #fff;     display: inline-block;     font-weight: 100;     text-align: center;     font-size: 20px; }  #clockdiv &gt; div{     padding: 5px;     border-radius: 3px;     background: #ff5b5b;     display: inline-block; }  #clockdiv div &gt; span{     padding: 5px;     border-radius: 3px;     background: #ab0202;     display: inline-block; }  .smalltext{     padding-top: 5px;     font-size: 13px; }  &lt;/style&gt;  &lt;div class="container"&gt;  &lt;h3&gt;Countdown Clock Left&lt;/h3&gt; &lt;div id="clockdiv"&gt;   &lt;div&gt;     &lt;span class="days"&gt;&lt;/span&gt;     &lt;div class="smalltext"&gt;Days&lt;/div&gt;   &lt;/div&gt;   &lt;div&gt;     &lt;span class="hours"&gt;&lt;/span&gt;     &lt;div class="smalltext"&gt;Hrs.&lt;/div&gt;   &lt;/div&gt;   &lt;div&gt;     &lt;span class="minutes"&gt;&lt;/span&gt;     &lt;div class="smalltext"&gt;Min.&lt;/div&gt;   &lt;/div&gt;   &lt;div&gt;     &lt;span class="seconds"&gt;&lt;/span&gt;     &lt;div class="smalltext"&gt;Sec.&lt;/div&gt;   &lt;/div&gt; &lt;/div&gt;&lt;/div&gt;  &lt;script&gt; function getTimeRemaining(endtime){   var t = Date.parse(endtime) - Date.parse(new Date());   var seconds = Math.floor( (t/1000) % 60 );   var minutes = Math.floor( (t/1000/60) % 60 );   var hours = Math.floor( (t/(1000*60*60)) % 24 );   var days = Math.floor( t/(1000*60*60*24) );   return {     'total': t,     'days': days,     'hours': hours,     'minutes': minutes,     'seconds': seconds   }; }  function initializeClock(id, endtime){   var clock = document.getElementById(id);   var daysSpan = clock.querySelector('.days');   var hoursSpan = clock.querySelector('.hours');   var minutesSpan = clock.querySelector('.minutes');   var secondsSpan = clock.querySelector('.seconds');    function updateClock(){     var t = getTimeRemaining(endtime);      daysSpan.innerHTML = t.days;     hoursSpan.innerHTML = ('0' + t.hours).slice(-2);     minutesSpan.innerHTML = ('0' + t.minutes).slice(-2);     secondsSpan.innerHTML = ('0' + t.seconds).slice(-2);      if(t.total&lt;=0){       clearInterval(timeinterval);     }   }    updateClock();   var timeinterval = setInterval(updateClock,1000); }  var deadline = 'October 31 2015 00:00:50'; initializeClock('clockdiv', deadline); &lt;/script&gt; I am trying to read a custom (non-standard) CSS property, set in a stylesheet (not the inline style attribute) and get its value. Take this CSS for example:  #someElement {   foo: 'bar'; }   I have managed to get its value with the currentStyle property in IE7:  var element = document.getElementById('someElement'); var val = element.currentStyle.foo;   But currentStyle is MS-specific. So I tried getComputedStyle() in Firefox 3 and Safari 3:  var val = getComputedStyle(element,null).foo;   ...and it returns undefined. Does anyone know a cross-browser way of retreiving a custom CSS property value?  (As you might have noticed, this isn't valid CSS. But it should work as long as the value follows the correct syntax. A better property name would be "-myNameSpace-foo" or something.) I've noticed that the majority of Ubuntu help websites present apt commands in the following format:  sudo add-apt-repository ppa:[ppa-address] sudo apt-get update sudo apt-get install [application-name]   But I know that apt commands like this one work too:  sudo add-apt-repository ppa:[ppa-address] &amp;&amp; sudo apt-get update &amp;&amp; sudo apt-get install [application-name]   Is there a particular reason that most sites present in the first format? Are there known issues with chaining commands with &amp;&amp;, or to put it another way are there any benefits/drawbacks to using either method?  Further, other then loss of internet, are there any circumstances in which an apt command might fail? Is it possible to find power armor in the wasteland? Why do we say "is it" when asking a question rather than "it is"? What it says on the tin, a foreign friend of mine has asked and I can't tell him; apart from it sounding horrible.  For example:  "Why is it raining today?"   Instead of:  "Why it is raining today?" In many languages, including English, the most important verbs are irregular. Examples include:   to be  to do  to get  to go  to have  to make   The same applies (roughly) to many other languages I know about (Dutch, German, French, Spanish, Swedish) and presumably to many other languages too.  Is there any reason why these everyday verbs tend to be irregular? At the end of a makefile I saw   rm -f *~ *.class   I understand the *.class, but what's *~? Will the Garbage Collector call IDisposable.Dispose for me?