The letter A appears an even number of times. How many different sequences could Dr. Lizardo have written down?
What is the number of strings at size $n$ that is constructed from ${a,b,c,d}$ and there is an even num of $a$
Changing color of already printed text For making a small quiz in class I am working on a presentation in beamer.  I want to change the color of statements to red (wrong statement) or green (right statement) but only after my class answered the question.  I was searching the internet, but only found the color package, where you apparently can only change the text color before it appears.  Can someone point me into the right direction? How can I add an application to the GNOME window manager? The GNOME on-board documentation covers only the things anyone can easily guess. I have an application for that I can only start from the command line. Not that I mind using a terminal but ...  Anyway, how do I add the command (and preferably a nice logo) to GNOME WM? "OpenID 2.0 for Google Accounts is going away"; will this affect Stack Exchange? Windows 8 removes Grub as default boot manager proving that elements of a sequence are pairwise coprime For the sequence $a_n = 6^{2^n} + 1$:   How can I prove that the elements of this sequence are pairwise coprime, i.e. prove that if $m$ is not equal to $n$ then $\gcd(a_m, a_n) = 1$. (begin by proving that $a_n∣ (a_{n+1} - 2)$) Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$?  Why? Thanks. After some searching, I find very little on the incorporation of observation weights/measurement errors into principal components analysis. What I do find tends to rely on iterative approaches to include weightings (e.g., ). My question is why is this approach necessary? Why can't we use the eigenvectors of the weighted covariance matrix? $dx$ appears in differential equations, such us derivatives and integrals.  For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is. How to make a circular array using offset object as center? The latest updates (4/1/2018) make my laptop constantly crash the GUI/Unity:  (Intel Pentium su4100 processor)  syslog:  Jan  5 11:38:53 1810 kernel: [ 1850.327738] compiz[11911]: segfault at 0 ip 00007f6c7baf0c16 sp 00007ffd5ec71c60 error 4 in i965_dri.so[7f6c7b513000+82d000]   After login I end up with an empty desktop, just a wallpaper, GUI/Unity are in a crash loop:   No panel.  No dash.  Can't start a program. Launcher disappears. Sidebar disappears. System crash. Windows key not working. The key combination Alt+Tab not working.   Simply, the computer is not usable. Same for guest login. The only thing working: open up a terminal CTRL+ALT+T to examine log files.  I somehow forced  to let me log in and file the bug, but still lots of problems:   Dash not working Alt+Tab not working Sometimes windows all disappear for a few seconds     Any ideas for a solution or workaround?   UPDATE     If you have the same/ very similar problems, follow the link to my    report and mark there that you're affected too. Or file your own. Please leave a comment   if you wish, or upvote - this is a nasty problem - but an answer should   offer a fix or a workaround so users can use their computer again.    Without , Ubuntu can't fix issues. Prove that lim $(\sqrt{n^2+n}-n) = \frac{1}{2}$ Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$  Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it spits out an error). Can anyone spot the error for me?  $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} &lt; \epsilon$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} &gt; \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \sqrt{n} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow n &gt; \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$ I have installed some apps through the Ubuntu Software center. How can I find the corresponding terminal command? For many apps, this is not the same as the displayed name. For example, the app called Text Editor can also be launched by typing "gedit" in the terminal, but how can I find this for other apps? When a user votes to close a question (for whatever reason), it would helpful to receive a notification (in the inbox) when the question is edited (either the text, tags, if it closes then is re-opened) so that the user can then modify his or her vote if the question is properly adjusted. Show that the function $g(x) = x^2 \sin(\frac{1}{x}) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$ Show that the function $g(x) = x^2 \sin\left(\frac{1}{x}\right) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$. Loading external GeoJSON file into Leaflet map? I would like to load a geoJSON (polygon) file into my leaflet map. I have seen examples where geoJSON is embedded into the javascript code but I can't find any examples showing how it is done with an external file.   &lt;head&gt;     &lt;meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no"&gt;     &lt;link rel="stylesheet" href="http://cdn.leafletjs.com/leaflet-0.5/leaflet.css" /&gt;     &lt;script src="usStates.geojson" type="text/javascript"&gt;&lt;/script&gt;     &lt;style&gt;         html, body, #map {             height: 100%;         }         body {             padding: 0;             margin: 0;         }     &lt;/style&gt; &lt;/head&gt;  &lt;body&gt;     &lt;div id="map" style="height: 100%"&lt;/div&gt;     &lt;script src="http://d3js.org/d3.v2.min.js?2.9.3"&gt;&lt;/script&gt;     &lt;script src="http://cdn.leafletjs.com/leaflet-0.5/leaflet.js"&gt;&lt;/script&gt;      &lt;script type="text/javascript"&gt;       var map = L.map('map').setView([38.57, -94.71], 4);      L.tileLayer('http://{s}.tile.cloudmade.com/9067860284bc491e92d2342cc51d47d9/998/256/{z}/{x}/{y}.png', {attribution: 'Map data &amp;copy; &lt;a href="http://openstreetmap.org"&gt;OpenStreetMap&lt;/a&gt; Imagery © &lt;a href="http://cloudmade.com"&gt;CloudMade&lt;/a&gt;'}).addTo(map);      var featureStyle = {         "color": "#ff7800",         "weight": 5,          "opacity": 0.2     };      L.geoJson(usStates).addTo(map);      &lt;/script&gt;  &lt;/body&gt; Adding Maximum and Minimum Latitude and Longitude as features in ArcGIS for Desktop? Blender is saving as a blend1 file, not blend I teach Blender to classes of game design students in the University of California and a student came to me yesterday with a problem I haven't seen before. She's created a very simple file, but when it's saved Blender puts a .blend1 extension on it every time, not the normal .blend. This means that after it's saved twice there are two .blend1 files in the directory, and no .blend file. I have no idea why this is happening. What has she done? Anyone seen this before?  thanks - Chris I have some HTML I am trying to parse.  There are cases where the html attributes alone are not going to help me identify the row type (header versus data).  Fortunately, if my row is a data row then it should have some values that can be converted to integers.  I have figured out how to convert the unicode to an integer for those cases that it is possible to make the conversion.  I am struggling to write the logic to move past the cells that the conversion will not work because the cell has content that must be treated as text.  for example if rowColumn[1][3] can be converted to an integer I can do so by   int(rowColumn[1][3].replace(',','').strip('$'))   but I get an error if rowColumn[1][3] has text content. How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in[1,2]^3$ Prove that for $x,y,z\in [1,2]$ the following inequality holds: $$\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}.$$ I tried to apply the Cauchy-Schwarz inequality or the power mean inequality, but my attempts were unsuccessfull, so I am asking for help. Why is asymmetric cryptography bad for huge data? I've been told that asymmetric cryptography requires that the message to be encrypted be smaller than its key length. Why is this?  I know about hybrid encryption, which . But I still want to know why public-key cryptography needs the data to be shorter than the key length.