Find volume of a translated and rotated ellipsoid below xy (horizontal) plane
Finding volume of a segment
Why does space expansion not expand matter? I have looked at other questions on this site (e.g. "why does space expansion affect matter") but can't find the answer I am looking for.  So here is my question: One often hears talk of space expanding when we talk about the speed of galaxies relative to ours. Why, if space is expanding, does matter not also expand? If a circle is drawn on balloon (2d plane), and the balloon expands, then the circle also expands. If matter is an object with 3 spatial dimensions, then when those 3 dimensions expand, so should the object.  If that was the case, we wouldn't see the universe as expanding at all, because we would be expanding (spatially) with it.  I have a few potential answers for this, which raise their own problems:   Fundamental particles are 'point sized' objects. They cannot expand because they do not have spatial dimension to begin with. The problem with this is that while the particles would not expand, the space between them would, leading to a point where the 3 non-gravity forces would no longer hold matter together due to distance Fundamental particles are curled up in additional dimensions a la string theory. These dimensions are not expanding. Same problems as 1, with the added problem of being a bit unsatisfying. The answer seems to be (from Marek in the previous question) that the gravitational force is so much weaker than the other forces that large (macro) objects move apart, but small (micro) objects stay together. However, this simple explanation seems to imply that expansion of space is a 'force' that can be overcome by a greater one. That doesn't sound right to me. Note: this is intended to be a canonical question about summoning sickness.  "Summoning sickness", broadly, is the rule that stops you from attacking or paying {T} costs with a creature that you just played. But how exactly does it work? How does it interact with other rules and effects?  For example, if I'm trying to figure out how a creature is affected:   Does the status matter (tapped/untapped, face down/face up, etc.)? What if it came into play, but I didn't cast it? What if it's a token? What if it was on the battlefield when my turn started, but it wasn't a creature? What if it stops being a creature? What if it's a creature and also another card type? What if it's a double-faced card that just transformed? What if it changes control? What if it's a copy of another creature? Which of the creature's activated abilities can I use? Are any of its other abilities affected? Can I tap the creature to pay a cost that is not a {T} cost?   (These examples are primarily taken from questions about summoning sickness that have already been asked on the site) More section headings? I am a novice in LaTeX and I have one question.  Can someone tell me what if you want more heading's in LaTeX. Like 5 or 10 or even 15. Is there a package for that? Book series about preparation for a war at the Singularity I am looking for a book series (almost certainly a trilogy) that starts with some humans in a space station being abducted by a small funny-looking alien and taken to a prison planet. They find out that the way space travel works is by creating copies of the passengers, and the originals remained on the space station. This is actually good for the aliens since they believe every sentient being will resurrect at the Singularity and will be forced to fight a battle, so the more the merrier.  The one other detail I remember is that the world economy has a rampant inflation, so people take any cash they have at the end of the day and buy 20th-century memorabilia that doesn't devalue as much. I'm writing a shell script for Linux, using Bash, to translate any video-file into a MP4. For that, I'm using avconv with libvorbis for audio.   Inside my script, I have a question for the user :   read -p "- Audio Quality [scale from -2 to 10] ? "     if [ -n "$REPLY" ] ; then     ABITRATE="-aq $REPLY"     fi   My "ABITRATE" string goes into the final avconv command-line.  But I would like to give the user the opportunity to answer that question with a value in Kb (Kilobit), and translate it into the scale that libvorbis uses. The "scale from -2 to 10" is this :   Quality Kbit/s  Normalization -----------------------------  -2      ~32        y  -1      ~48        y   0      ~64        y   1      ~80        y   2      ~96        y   3     ~112        y   4     ~128        n   5     ~160        n   6     ~192        n   7     ~224        n   8     ~256        n   9     ~320        n  10     ~500        n   I would like to know how to check if my $REPLY is in a range of number. For example, I would like my script to do something like this :   if [ $REPLY is a number between 1 and 32 ] ; then   REPLY="-2" elif [ $REPLY is a number between 33 and 48 ] ; then   REPLY="-1" fi   Is this possible (I'm willing to say 'yes of course, shouldn't be hard' but I don't know the syntax to use) ? Arithmetic of continued fractions, does it exist? I'm interested in the arithmmetic of continued fractions and specially in multiplication. Consider  $$ f(x)=\cfrac{f_{0}(x)}{1-\cfrac{f_{1}(x)}{1+f_{1}(x)-\cfrac{f_{2}(x)}{1+f_{2}(x)-\cfrac{f_{3}(x)}{1+f_{3}(x)-\cdots}}}} $$   and   $$ g(x)=\cfrac{g_{0}(x)}{1-\cfrac{g_{1}(x)}{1+g_{1}(x)-\cfrac{g_{2}(x)}{1+g_{2}(x)-\cfrac{g_{3}(x)}{1+g_{3}(x)-\cdots}}}} $$   Are there arithmetic rules (algorithms) for the multiplication of continued fractions? Specifically, is it possible to obtain a continued fraction $h(x)=f(x)\cdot g(x)$ where   $$ h(x)=\cfrac{h_{0}(x)}{1-\cfrac{h_{1}(x)}{1+h_{1}(x)-\cfrac{h_{2}(x)}{1+h_{2}(x)-\cfrac{h_{3}(x)}{1+h_{3}(x)-\cdots}}}} $$   I've found , but I'd like more. Does anyone here knows of papers, algorithms, etc?  Thanks.    Update 1: Another . What's the best way to protect your house from Endermen stealing your walls? It shows no question owner, not even a simple name (without link).   What happened? My chooser (file-picker dialog) goes under the dock so I can't see buttons. What can I do? Programmatically disable apt unattended upgrades How can I programmatically/script disable apt unattended upgrades? Running Ubuntu 16.04. What does "T" mean in C#? I have a VB background and I'm converting to C# for my new job.  I'm also trying to get better at .NET in general.  I've seen the keyword "T" used a lot in samples people post.  What does the "T" mean in C#?  For example:  public class SomeBase&lt;T&gt; where T : SomeBase&lt;T&gt;, new()   What does T do? Why would I want to use it? How would you explain covariance to someone who understands only the mean? ...assuming that I'm able to augment their knowledge about variance in an intuitive fashion (  ) or by saying: It's the average distance of the data values from the 'mean' - and since variance is in square units, we take the square root to keep the units same and that is called standard deviation.  Let's assume this much is articulated and (hopefully) understood by the 'receiver'. Now what is covariance and how would one explain it in simple English without the use of any mathematical terms/formulae? (I.e., intuitive explanation. ;)  Please note: I do know the formulae and the math behind the concept. I want to be able to 'explain' the same in an easy to understand fashion, without including the math; i.e., what does 'covariance' even mean? If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational Assume there exist some rationals $a, b$ such that $\sqrt[3]{a}, \sqrt[3]{b}$ are irrationals, but:  $$\sqrt[3]{a} + \sqrt[3]{b} = \frac{m}{n}$$  for some integers $m, n$  $$\implies \left(\sqrt[3]{a} + \sqrt[3]{b}\right)^3 = \frac{m^3}{n^3}$$  $$\implies a + b + 3 \cdot \sqrt[3]{ab}\left(\sqrt[3]{a} + \sqrt[3]{b}\right) = \frac{m^3}{n^3}$$  Since $a +b$, $\sqrt[3]{a} + \sqrt[3]{b}$ are rational, $\sqrt[3]{ab}$ must be rational as well.   For convenience let us say $\sqrt[3]{a} = p, \sqrt[3]{b} = q \implies pq$ is rational. This means, for all $i$, $p^iq^i$ is rational.  $$\implies (p + q)^2 = \frac{m^2}{n^2}$$  $$ = p^2 + q^2 + 2pq = \frac{m^2}{n^2}$$  Since $pq$ is rational, $2pq$ is rational and so is $p^2 + q^2$.  Assume, for some $i$ that $p^i + q^i$ and $p^{i-1} + q^{i-1}$ is rational.   $$\implies (p^i + q^i)(p + q) - pq(p^{i - 1} + q^{i - 1}) = p^{i+1} + q^{i+1}$$  is rational as well.  So for all $i$,  $$a^{\frac{i}{3}} + b^{\frac{i}{3}}$$  and   $$a^{\frac{i}{3}}b^{\frac{i}{3}}$$  are rational.  I know I'm really close to the answer, but it somehow just keeps slipping through my fingers. The coefficient of an explanatory variable in a multiple regression tells us the relationship of that explanatory variable with the dependent variable. All this, while 'controlling' for the other explanatory variables.  How I have viewed it so far:  While each coefficient is being calculated, the other variables are not taken into account, so I consider them to be ignored.   So am I right when I think that the terms 'controlled' and 'ignored' can be used interchangeably? Is there a subtle difference between "somebody" and "someone", "anybody" and "anyone"? Are there any subtle differences between "somebody" and "someone", or can they be used completely interchangeably? Similarly, can you imagine a situation in which you would prefer "anybody" to "anyone" or vice versa? Let $X_n$ be iid random variables with $P(X_n=(-1)^kk)=\dfrac{C}{k^2\log k}$ for all $k\geq 2$. Here $C$ is just a constant so that the sum of probabilities is $1$. Find constant $\mu$ such that $S_n/n\to \mu$ in probability where $S_n=\sum_{k=1}^nX_k$.   Here $E(X_n)=\infty$. I'm a bit lost after this. In general how does one find constants like this? moderncv: How to control the length of cvitems and cvlines? I would have expected the correlation coefficient to be the same as a regression slope (beta), however having just compared the two, they are different. How do they differ - what different information do they give? Is time continuous or discrete? Contains and Custom Label in Rule Criteria - Workflow I have a picklist field-XYZ which contains a lot of values.Lets say (A,b,c,d...z) I am trying to write a rule criteria which will execute if the picklist value is any of the following (A,b,c...p).So instead of mentioning all the values in rule criteria , i created a custom label and put the values (a,b,c,d..p) in that and wrote the following in the rule criteria :   CONTAINS(TEXT(XYZ__c) ,$Label.CustomLabel))   This is not working even is the value of the picklist is equal to the one mentioned in the Custom Label.I guess the CONTAINS function is matching the whole value of custom label with Picklist.Is there other workaround/solution for this?