Triangles are used in structures because they are the most rigid shape. And they are many descriptions of this on network and, if I am not mistaken, this all comes down to the law of cosines.But why this holds? Is it maybe because if we connect three random points on a planes their internal angles will add to half revolution? Is there a more fundamental reason for the rigidness of triangles? Engineer here, please don't mind the naive question
does not appear to actually define what a rigid shape is. Rather it defines 'same shape' and 'rigid transformations' without giving any definitions of what is necessary and sufficient for a shape to be considered rigid.  For instance, I've seen the following image:    I understand intuitively why the triangle is rigid and quadrilateral is non-rigid. It is also my understanding that inserting a single diagonal connection into the quadrilateral would make the shape rigid? However, given the following image (labeling the shapes $S_1$, $S_2$ and $S_3$ in sequence left to right):    If $S_1$ had a single diagonal connection AB, it would be possible to flip ADB over axis q to obtain a shape similar to $S_2$ with the addition of connection AB (my explanation may not be the greatest). This shape is clearly not the same as $S_1$, so how can we say $S_1$ with a single diagonal connection CD is rigid? I feel I am unclear on what the definition of rigidity actually is.
When and where does Attack on Titan take place? Is the setting of "Attack on Titan" (Shingeki no kyojin) supposed to be Earth in the distant future and Titans suddenly appeared out of nowhere and devolved society to medieval level of technology? Or is this just a completely fictional setting? What is a mixin, and why are they useful? I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values with a calculator in order to determine which of the two is larger.  Can anyone demonstrate an algebraic/analytic method to find out which of the two is larger without using a calculator? Let $V$ be a representation with character $\chi$.  I would like to have a formula for the characters of the representations $\mathrm{Sym}^m[V]$ and $\wedge ^m[V]$ in terms of $\chi$.  Fulton and Harris presents such a formula for $m=2$, but not for general $m$.  Is there a "nice" such expression?  I found a paper by , but this doesn't seem to be what I'm looking for (it doesn't seem to give a general formula in terms of $\chi$).  In case the link does not work, here is the reference:  X. Zhou, P. Pulay. Characters for Symmetric and Antisymmetric Higher Powers of Representations: Application to  the  Number of Anharmonic Force Constants in Symmetrical Molecules. Journal of Computational Chemistry, Vol 10, Issue 7, (1989). I understand if I boot from a live cd  I can see all the system logs under System > Administration > File Log Viewer   I have a major error with a disk not mounting and I want to trace i tback to the last time it did work and what may have corrupted the ext4 filesystem on it.  So within the File Log Viewer Where do I start examining? How do I bake multiple texture maps into 1 UV, In Blender Cycles? Using:  Blender V2.79 Cycles Render Engine  After creating a complex interior scene consisting of 3 different texture types, each having 3 associated maps (Diffuse, Gloss, Normal) I noticed the texture and baking options will not successfully combine to create a texture for the exported mesh once I bring it into Unity 3D. The mesh is a single object with the different textures assigned to specified faces.  I've played around with a default cube mesh for testing purposes, using the same 3 textures split among the 6 sides, but have only been able to successfully pull over the color info by baking a diffuse map to my UV and saving the image. When using the other baking options, Blender does not successfully create a map for either my Glossy textures, nor my normal map. Even using the "Combined" baking option, I am provided a flat dull diffuse image for my mesh, no gloss or normal info is saved.  My Node process for textures is displayed here  Here is the result of baking my textures as a Normal  The result of adding other textures made from Blender's baking, results in flat gloss at best.  What can I do differently in Blender, to have my assigned UV Diffuse, Normal, and Gloss maps all appear in a single object in Unity? Finding the limit of roots products $(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2})$ Number of isosceles triangle. I have a cron job that runs every minute on an Ubuntu 12.04 box  $sudo crontab -e * * * * * mylogin /pathto/file.sh &gt; /log/filelog   If I run file.sh, the bash script does some stuff and echos out runs:  $ ./file.sh  runs   If I check the log of cron tab it shows that the job is running:  Jul 10 12:41:01 localhost CRON[1811]: (root) CMD (mylogin /pathto/file.sh &gt; /log/filelog) Jul 10 12:41:01 localhost CRON[1810]: (CRON) info (No MTA installed, discarding output) Jul 10 12:42:01 localhost CRON[1813]: (root) CMD (mylogin /pathto/file.sh &gt; /log/filelog)   However, the script is not running. It is not doing it's work and not echoing runs to /log/filelog.  $cat /log/filelog  #shows nothing   What other steps can I take to debug this problem? Why is ROC AUC equivalent to the probability that two randomly-selected samples are correctly ranked? I found there are two ways to understand what AUC stands for but I couldn't get why these two interpretations are equivalent mathematically.   In the first interpretation, AUC is the area under the ROC curve. Picking points from 0 to 1 as threshold and calculate sensitivity and specificity accordingly. When we plot them against each other, we get ROC curve.  The second one is that the AUC of a classifier is equal to the probability that the classifier will rank a randomly chosen positive example higher than a randomly chosen negative example, i.e. P(score(x+)>score(x−)).  (from ) Looking at the properties for a Windows file I get two attributes, “Size” and “Size on disk,”  and "Size on disk" is always larger.  What do these two metrics mean? What is the weight equation through general relativity? The gravitational force on your body, called your weight, pushes you down onto the floor. $$W=mg$$ So, what is the weight equation through general relativity? TV series with pull out phone screens and pink/purple space ships I’m looking for a TV show where an alien race comes to Earth and sets up big purple/pink cylindrical or cone shaped spaceships in various cities around the world. The Aliens are not hostile but actually created peace and eliminated hunger on Earth.    I believe the aliens had some human features and were wise and calm the main actor had a phone with a pull out screen I believe the main actor worked for or with the aliens I believe that in the last episode some humans and aliens go into some kind of pods and perhaps leave Earth? one of the space ships was located in a major US city. Maybe Washington DC or another East Coast city   Could someone please tell me what TV series this is? I have been searching everywhere and cannot find it. Thanks! How can I disable the Windows 8 charms bar? Can someone please tell me how to disable the annoying effect that pops up when you drag cursor to top- or bottom-right corners? bash: adduser: command not found on centOS I installed CentOS 5.5 on my VMWare 8 recently and I am trying to add a new user on the system. I am unable to add the user unless I use su - option. I believe it has to do something with path not set properly.  I updated the path and here is what it looks like  /usr/kerberos/sbin:/usr/kerberos/bin:/usr/local/bin:/usr/bin:/bin:/usr/X11R6/bin:/home/uone/bin:/sbin   I believe the command is in /sbin dir which is already a part of path. Can anyone suggest me what else I might be missing? What happens when anti-matter falls into a black hole? Let's say a black hole of mass $M$ and a very compact lump of anti-matter () also of mass $M$ are traveling toward each other. What does an outside observer see when they meet?  Will they blow themselves apart in a  matter/anti-matter reaction? Or will their masses combine, never quite meeting in the infinite time dilation at the event horizon? How to prove that there exist no functions $f,g:\Bbb{R}\to\Bbb{R}$ such that $f(g(x))=x^{2018}$ and $g(f(x))=x^{2019}$? I have tried a little bit to solve the problem which goes as follows:   My intuition says that there exist no $f:\Bbb R\to\Bbb R$ such that $$f(g(x))=x^{2018}\text{ and }g(f(x))=x^{2019}.$$  Note that $$f(g(f(x)))=f(x)^{2018}\implies f(x^{2019})=f(x)^{2018}$$  Similarly, $$g(x^{2018})=g(x)^{2019}$$ Putting $x=1$ in $f(x^{2019})=f(x)^{2018}$, we get $f(1)=f(1)^{2018}$ and thus $f(1)=0$ or $f(1)=1$.  Similarly, putting $x=1$ in $g(x^{2018})=g(x)^{2019}$ we get $g(1)=g(1)^{2019}$ and thus $g(1)=0,1,-1$.  Now, I can't proceed further. Can anybody solve it? Thanks for assistance in advance. I resized the root partition with GParted, from 20 GB to 40 GB, but the filesystem still reports a size of 20 GB.  How can I use all 40 GB?    Update  The resized partition was an LVM Physical Volume (/dev/sda2), and the boot partition is ext4 (/dev/sda1). Website logo text - best tag for SEO purposes? I'm trying to develop a new website in HTML5 adhering to SEO standards where possible and I'm having a little trouble trying to figure out what tag my site name and slogan should live in.  Originally I had my site name in a &lt;h1&gt; tag and my sloagan in a &lt;h2&gt;, but soon realised this wasn't fitting with the "each page is a separate document/article" approach.  The site name is the largest text on the page (sometimes the same as my article heading), but I've currently got the text in a &lt;p&gt; tag with CSS doing all the styling work.  My questions is, what should I do with the site name and slogan? Is the &lt;h1&gt; appropriate, then use another &lt;h1&gt; tag for each page title or should I continue using the styled &lt;p&gt; tag? Or should I convert this into an image and use that for the site name/slogan?  (Side note: on my index page I want the site name and slogan to be &lt;h1&gt; and &lt;h2&gt;respectively as this makes sense with the homepage...) ConcurrentModificationException for ArrayList