Does $m^*(E)=m(E\cap A)+m^*(E\cap A^c)$ always true?
Intuition behind the definition of a measurable set
How to solve this probability problem analytically (instead of using simulation)? Probability of Bill wins the game if he goes first "In a party people shake hands with one another" In a party people shake hands with one another (not necessarily everyone with everyone else).   (a) Show that 2 people shake hands the same no. of times.   (b) Show that the number of people who shake hands an odd no. of time is even.        I find this question to be very tough and I can't find a way to even start answering this. It would be great if I could get a small hint in order for me to at least start it. Sublime Text 3 System Package Location? Calendar.getInstance(TimeZone.getTimeZone("UTC")) is not returning UTC time Proof that the Euler characteristic is additive I'm reading through a set of notes which assumes that the Euler characteristic is additive, but doesn't give a proof, so I would like to understand why this is.  Let $A_n$ be a finitely generated Abelian group, non-zero for $0&lt;n&lt;k$. Consider the long exact sequence  $ \cdots \rightarrow A_n \rightarrow B_n \rightarrow C_n \rightarrow A_{n-1} \rightarrow B_{n-1} \rightarrow C_{n-1} \rightarrow \cdots $  where $B_n$, $C_n$ defined similarly to $A_n$.   I know that the Euler characteristic of $A$ is given by $\chi(A) = \sum_0^k (-1)^n \text{rank}(A_n)$, with similar definitions for $B$ and $C$. I want to prove that $\chi(B) = \chi(A) + \chi(C)$.  The notes say that for an exact sequence  $0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0$  of finitely generated Abelian groups $X,Y,Z$, we have the result rank($Y$) $=$ rank($X$) +  rank($Z$), and that the additivity of the Euler characteristic follows from this fact, but I can't see how. I have an armature set up, but the bones aren't moving the mesh properly  The mesh and armature:    The vertex group of the bone I am going to move:    As you can see, the circled region doesn't belong to the vertex group. But, when I move the bone look what happens:    That part of the mesh has moved with the bone.. Why? Poiseuille's law states that the rate of flow of water is proportionate to $r^4$ where $r$ is the radius of the pipe. I don't see why.  Intuitively I would expect rate of flow of fluid to vary with $r^2$ as the volume of a cylinder varies with $r^2$ for a constant length. The volume that flows past a point is equal to rate of flow fluid, and thus it should vary with $r^2$.  I cannot understand the mathematical proof completely. Is there an intuitive explanation for this? Probability of student passing an exam There're $25$ tests, each with $2$ questions. There're total of $50$ different questions (the questions do not repeat). The student knows the answer of $44$ questions. To pass the exam, the student needs to correctly answer both questions on the first test he drew, or answer $1$ question from the test he drew first, and $1$ question from the test he drew second. (meaning he draws twice if he answers a question incorrectly on his first try)  What is the probability that the student will pass the exam?  I have never faced a problem like this one so any help is appreciated. I was thinking about approaching it "manually" with some applied combinatorics but I feel like there's a much easier way that I am not familiar with. Where are the Unity desktop wallpapers located? Please see the picture below. Where are the Unity desktop wallpapers located? I recently installed Ubuntu 14.04, then I installed lamp-server and placed my webpages in /var/www directory, but when I opened localhost in the browser there was nothing. I think that happened because Zend updated Apache.  Whatever the reason may be, I want to know where to keep my php files so that I can access them from my browser. If you shine a light away from the center of a Schwarzchild black hole, will it stay still? $$ A $$  Light is travelling away from the absolute center of the Schwarzchild black hole (i.e. none of that spinning/stretching field) at $\ C $, and so shouldn't $\ \Delta C$ need to be greater than $\ C $ in order for the light to collapse into the black hole, therefore violating the relativistic limit of $\ C $?    It seems like light would just 'hover' right in the center of a black hole then. Because if it goes away from it, the event horizon pulls it back, and if it goes into it, the velocity change must be greater than $\ C $ which violates general relativity; so technically there's no way for the light to move away, and so it would just be still.  The thing is that the light must move at the speed of light or general relativity doesn't work. It's massless... But in the center of a black hole, if the above is true (it can't move or escape), then it is not travelling at the speed of light. So it cannot be still.  It cannot be spinning in a constant orbit either, because that would require velocity in an outward angle. And it cannot move in an outwardly fashion, only in an inwardly one.  Is it spinning in an infinitely decaying spiral towards the center?  If so what happens to it if and when it reaches the direct and complete center, i.e. The case of it being 'shot' from the center.    $$ B $$  If you shot light at an angle towards a black hole, but not directly to the center of it  then that light from the outside will eventually become the same as the light that I said was shot from the center to the outside; because in both cases, they end up in an infinitely decaying spiral.    If so it would end up exactly like the light in the above paragraph, so then we need to figure out what happens to that light to figure this one out, or vice versa.    $$ C $$  If you shot light directly towards the center of a black hole it would have no spiral to decay into, so therefore it must stop at some point in the center of a black hole. If it stops though, it loses it's momentum and violates general relativity again because it's energy "disappears".    If so then it violates conservation laws. So I'm assuming this simply is not true.    Is the light travelling into a black hole somehow still travelling at the speed of light, in every direction possible where that direction keeps it facing towards the center?  If so, why is there not a point at which the only thing left for the light to do is stay still, since all other paths would increase the distance between it and the black hole's center, and therefore become Case A. And if there isn't, what would happen if there was?  Also, if it's a time and observation answer where it supposedly takes and infinite amount of time for C to hit the center, wouldn't it be the case that the black hole is 'eating' all of the available light in the universe and keeping it in stasis?  Also, what happens in a Kerr black hole then? Does it just rip the field of time/gravity at an angle such that the light would never get closer than the center? Why does a wireless network give a more accurate location than a wired network? I build a simple HTML5 website that takes my location (in latitude-longitude), and simply shows my location in Google maps.  I opened the HTML document with my laptop Asus K53E-B1 and I found that when I connected to a wireless network, it showed almost my precise location (about 100 meter of inaccuracy),  while when I plug in the network cable, and test again, it's a bit Inaccurate. (showed a street 12KM away from me).  Can you explain me why is a wireless network gives more accurate location?  Tested with Google Chrome. Problem: If   $$C_0+\frac{C_1}{2}+\cdots + \frac{C_{n-1}}{n}+\frac{C_n}{n+1} =0,$$  where $C_0,...,C_n$ are real constants, prove that the equation  $$C_0+C_1x+\cdots +C_{n-1}x^{n-1}+C_nx^n=0$$  has at least one real root between $0$ and $1$.  Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 4. I would like to know if anyone can help me with installing Ubuntu on VirtualBox Can I manually put an external hard disk to sleep? I need to delete some rows from one large table. The rows to delete shouldn't be in another table, example:  DELETE FROM LargeTable WHERE id IS NOT IN (SELECT DISTINCT foreign_id from EvenLargerTable)  but my server can't handle such blunt query, because there are almost a million records in the LargeTable and few million records in the EvenLargerTable  How can I solve it? How to fix lsregister failing associations after reboot? Can a full version of Ubuntu be Installed on a USB Drive? Allow the author to delete their own question after closure, if other rules permit it I've read through a few "I want to delete my own question" threads on  already, and haven't found any that particularly cover this case.  I asked a question recently on :    This was voted closed as a duplicate of this question:    Since the latter question was obviously an appropriate duplicate, and my question had no votes or answers whatsoever, I decided I would just delete it.  But then, I was greeted by this error:     To allow for possible reopening, you may delete in 2 days     This all happened within hardly 20 minutes of me posting the question in the first place.  I fail to understand the point of this.  Perhaps if there were some up-votes or answers to the question, it might make sense.  However, as it stands, we now have a question with:   No votes whatsoever (up or down). No answers. Only a single (and largely irrelevant) comment not made by the OP. Five 3k+ rep users agreeing it is duplicate and therefore superfluous. The OP expressly desiring the question to be deleted, implying:   The OP has probably reviewed the duplicate question, and found their answer there. The question holds no further value to the OP. The OP agrees (with the close-voters) that the question provides no added value to the community.    At this point, what good reason could there be for disallowing the OP to delete their own question?  Could the system be modified to permit this? How can we distribute Shamir's secret sharing scheme shares without a dealer? Let's say $k$ users have shares to a $(k,n)$ secret sharing scheme, and they want to issue new shares. How can they do this without collecting the shares together?  In particular, no set of $k-1$ users (including the new users) should be able to learn the shared secret during or after the share generating process. There is also no trusted third party.  One way to do this is by doing a secure multiparty computation. Is there a way to accomplish the task without a general multiparty computation scheme?