Discs are naturally occurring shapes in the world, because any soft object experiencing rotation on a single axis will stretch itself into a disc shape such as pizzas and galaxies due to centrifugal forces caused by rotation. Galaxies and pizzas are "soft" in the sense that the tensile forces holding the structure together are not strong enough to resists being stretched by rotation. Rocks on the other hand, do not stretch when rotating because the average tensile forces between the atoms of a rock are strong enough to resist centrifugal forces, given that rotation is not too fast. If Earth is mostly fluid, why does it not stretch as it rotates?
The question I am working on is, &quot;Consider the following. (a) Find the angular speed of Earth's rotation about its axis. rad/s (b) How does this rotation affect the shape of Earth?&quot;  I am fully capable of solving part (a); however, I am not sure how to describe the effect earth's rotation on its shape. I tried to search my textbook for the answer, but could not find anything. Is there an actual effect on the shape?
Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$. Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2}  \right |\leq \left |b-c  \right |$. Give me a clue, I don't have any idea to work with that inequality. I am trying to rotate a dual monitor which is attached to the VGA output of my laptop. My system is Ubuntu 15.10.  However, here is the outcome I get when I use xrandr:  xrandr --output VGA-1-0 --rotate right xrandr: output VGA-1-0 cannot use rotation "right" reflection "none"   I have seen other posts on this subject, which all recommend editing xorg.conf. I have tried that, too. Here's my current xorg.conf file:  Section "ServerLayout" Identifier "layout" Screen 0 "nvidia" Inactive "intel" EndSection  Section "Device" Identifier "intel" Driver "modesetting" BusID "PCI:0@0:2:0" Option "AccelMethod" "None" Option "RandRRotation" "True" EndSection  Section "Screen" Identifier "intel" Device "intel" EndSection  Section "Device" Identifier "nvidia" Driver "nvidia" BusID "PCI:1@0:0:0" Option "ConstrainCursor" "off" Option "RandRRotation" "True" EndSection  Section "Screen" Identifier "nvidia" Device "nvidia" Option "AllowEmptyInitialConfiguration" "on" Option "IgnoreDisplayDevices" "CRT" Option "RandRRotation" "True" EndSection   Still, I keep getting the same error when trying to use xrandr to rotate that screen.   Is there another way of doing so? Blender build for Raspberry Pi3? SHOWPLAN permission denied when including execution plan in Data Explorer If I select the "Include execution plan" checkbox in the Data Explorer, the query fails:     SHOWPLAN permission denied in database 'StackOverflow'.     I see that  following maintenance. I don't know whether there was any recent maintenance. It's been several days since the data was updated and longer since an update was deployed, if that's related. If I have Optimize iPhone Storage enabled on my iPhone and my phone storage is completely full, I understand that iOS will only store a low-res thumbnail of many of my photos on the device. (The full high-res version of the image will only exist in iCloud and on devices with Download and Keep Originals enabled.)  But when only thumbnails exist on the device, how does that work in relation to other photo apps on iOS? For example, do third-party cloud-based photo apps (like Google Photos, Dropbox's Carousel, or Amazon Prime Photos) have access to the full high-res images from the iCloud Photo Library? Or will they only see the low-res thumbnails in some cases? How to find perpendicular vector to another vector? How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please?  I have a solution to this when I have $3\mathbf{i}+4\mathbf{j}$, but could not solve if I have $3$ components...  When I googled, I saw the direct solution but did not find a process or method to follow. Kindly let me know the way to do it. Thanks. Find and Replace several several different values all at once Using conjugates to find a limit with a cubic root: $\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$ I currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$  Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards. If it were a sqrt I know that the limit is 1/2. I know that the result here is 1/3 but I can't seem to get there. I always end up with 1/4 due to getting something like: $$\frac{h+1-1}{h(\sqrt[3]{h+1}+1)(\sqrt[3]{h+1}+1)}$$  After simplifying and evaluating the limit you end up with $$\frac{1}{(\sqrt[3]{0+1}+1)(\sqrt[3]{0+1}+1)}$$  which turns out to be $1/4$.  What am I doing wrong? How can I prove the convergence of the sequence $b_n=\sqrt[n]{x^n+y^n}$ where $x, y &gt; 0$?  Can I divide it in two cases?  Case 1: $x &gt; y$.  $$ b_n=\sqrt[n]{x^n+y^n} &lt; \sqrt[n]{x^n+x^n} = \sqrt[n]{2 \cdot x^n}=x \cdot \sqrt[n]{2} $$  Case 2: $x &lt; y$.  $$ b_n=\sqrt[n]{x^n+y^n} &lt; \sqrt[n]{y^n+y^n} = \sqrt[n]{2 \cdot y^n}=y \cdot \sqrt[n]{2}$$  Result: Does the sequence converges to $\max(x,y)$? Logistic regression in R resulted in perfect separation (Hauck-Donner phenomenon). Now what? I'm trying to predict a binary outcome using 50 continuous explanatory variables (the range of most of the variables is $-\infty$ to $\infty$). My data set has almost 24,000 rows. When I run glm in R, I get:  Warning messages:   1: glm.fit: algorithm did not converge   2: glm.fit: fitted probabilities numerically 0 or 1 occurred    I've read the other responses that suggest perfect separation might be occurring, but I'm confident that isn't the case in my data (though quasi-complete separation could exist; how can I test to see if that's the case?). If I remove some variables, the "did not converge" error might go away. But that's not always what happens.  I tried using the same variables in a bayesglm function and got the same errors.  What steps would you take to figure out exactly what's going on here? How do you figure out which variables are causing the problems? I am solving a clustering problem on a trivial dataset with the k-means algorithm. I am running a couple of tests and, from time to time, one centroid doesn't attract any points, i.e. I've got an empty cluster (see the purple "x" in the picture).  What should I do? Shall I delete it or just stop updating its value? Why?    I am aware that built-in functions (e.g., kmeans() in R) have automatic ways of dealing with this situation, but I am trying to write the standard algorithm from scratch. As soon as I fix it I'll be able to compare my results to built-in functions. At this moment I'm looking for some theoretical reasons why I should prefer one solution or another.  In the picture each colour represents a cluster according to the current iteration and each "X" is its centroid (old ones have been kept and marked with the number of the iteration in which they were computed).      [Some related links about empty cluster case, added later by @ttnphns: ; ; ; ; ; ; ; ; ] How do I access previous promise results in a .then() chain? What if residuals are normally distributed, but y is not? I've got a weird question. Assume that you have a small sample where the dependent variable that you're going to analyze with a simple linear model is highly left skewed. Thus you assume that $u$ is not normally distributed, because this would result in normally distributed $y$. But when you compute the QQ-Normal plot there is evidence, that the residuals are normally distributed. Thus anyone can assume that the error term is normally distributed, although $y$ is not. So what does it mean, when the error term seems to be normally distributed, but $y$ does not? Q) Prove that if $0\rightarrow V_1\rightarrow V_2\rightarrow...\rightarrow V_k\rightarrow 0$ is an exact sequence of finite dimensional vector spaces, then $\sum_{i=1}^{k}(-1)^idim (V_i) = 0$.  For $k=3$, I see that the statement is true. For $k&gt;3$, am I correct in understanding that $V_2 = V_1\oplus V_3 (\implies dim(V_2)=dim(V_1)+dim(V_3))$ and $V_3 = V_2\oplus V_4(\implies dim(V_3)=dim(V_2)+dim(V_4))$ and so on?   If it is true, then for $k=4$, $\sum_{i=1}^{4}(-1)^idim (V_i) = -m_1+m_1+m_3-m_2-m_4+m_4$ and am not sure why $m_2=m_3$ for the sum to be $0$? where $m_i = dim(V_i)$. Thanks. What is the function of "do" in this sentence? Rarely, though, do countries introduce a single reform alone.   What function does do have in this sentence? How to adjust the gap between figure and caption within the same subfloat Restoring Windows XP PATH variable I installed an application that screwed my PATH variable. Now I cannot run anything from the command prompt. Is there any way I can restore PATH variable? $\sum_{n=1}^\infty \frac{i^n}{n}$ converge? Dealing with multiple private DNS servers A very easy probability problem but opinions diverge greatly. Need some help. The question itself is a very easy one:     Somebody has got two kids, one of whom is a girl. Then what's the probability that he's got at least one boy?   My answer is that, since he's already got a girl, then "he's got at least one boy" amounts to "the other kid is a boy", whose probability is apparently $\frac{1}{2}$. But my friends argue that the probability should be $\frac23$: they say this is a binomial distribution, all the possible cases are (girl,girl),(girl,boy),(boy,girl) which yields that the probability is two cases out of three and is thus $\frac23$. But I think this is totally unacceptable. I don't think it is a binomial distribution at all, at least not what my friends explained to me. However, I just can't disuade them of their opinion, nor can I prove that I am wrong. So what on earth is the probability? and why? Any help is appreciated. Thanks in advance. Esp. Can anybody show why my explanation is wrong? Isn't it that whether the other kid is a boy or a girl a 50/50 event?  EDIT: Thanks for all the help you provided for me, and special thanks will go to @HammyTheGreek and @KSmarts, who have made it clear to me that there is in fact some ambiguity in my statement in this problem. As is pointed in  ,two distinct interpretation of the statement "one of whom is a girl" that gives rise to ambiguity:     From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.   From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of 1/2.