Pairwise independence of three events does not imply their mutual independence.How?
What is an example of pairwise independent random variables which are not independent?
I have a document divided into \part \chapter \section and \subsection. I have 3 parts and in each part I reset all the counters.  I want that, when I do a crossref to something that is in the 1st part, chapter 3 and section 5, that LaTeX prints something like I-3.5, but  I don't want that in the tableofcontents part that LaTeX prints the part number. For example, what I want is:     Part I      chapter 1 section 1.1 section 1.2      chapter 2   AND NOT     chapter I-1 section I-1.1 section I-1.2      chapter I-2   The command that I've tried   \renewcommand{\thechapter}{\thepart-\arabic{chapter}} \renewcommand{\thesection}{\thechapter.\arabic{section}}   does exactly what I've said  Can anyone help me? Exercise 3.5.12. Let $ f: \mathbb N \times \mathbb N \to \mathbb N$ be a function, and let c be a natural number. Show that there exists a function $a:\mathbb N\to \mathbb N$ such that $$a(0)=c$$ and $$a(n++)=f(n,a(n))\;for\;all\;n\in \mathbb N,$$ and furthermore that this function is unique. Tao gives the hint: first show inductively, by a modification of the proof of Lemma 3.5.12, that for every natural number $N\in \mathbb N$, there exists a unique function $a_N:\{ n\in \mathbb N:n\lt N\}\to \mathbb N $ such that $a_N(0)=c$ and $ a_N(n++)=f(n,a(n))$ for all $n\in \mathbb N$ such that $n\lt N$.  In my opinion, we could solve it by proving that $\forall n\in \mathbb N,a(n)\in \mathbb N$ according to mathematcial induction.  The property P(n) is:$$\forall n\in \mathbb N, a(n)\;is\;unique$$ P(0) is true. And if P(n) holds, P(n++) is true because different natural numbers must have different successors. $$a(n++)=f(n,a(n))$$ $$a(m++)=f(m,a(m))$$ $$n++=m++\Rightarrow n=m\Rightarrow a(n++)=a(m++)$$ It's similar to the proof of  Proposition 2.1.16 (Recursive definitions): Suppose for each natural number $n$, we have some function $f_{n} : \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to the natural numbers. Let $c$ be a natural number. Then we can assign a unique natural number $a_{n}$ to each natural number $n$, such that $a_{0} = c$ and $a_{n{++}} = f_{n} (a_{n})$ for each natural number $n$.  This proof indicates both the existence and uniqueness of the function $a$. I don't know why Tao says it's informal.  But regarding the hint, how do we get the function f even if we've got the conclusion from the hint. Also it seems like the function a(n) can be diffentent considering different N. I don't know if the hint is correct. Energy conservation in General Relativity I understand that energy conservation is not a rule in general relativity, but I'd like to know under what circumstances it can still be possible. In other words, when is it possible to associate a potential energy to the gravitational field, so that the energy is constant in the evolution of the system?  Here are some examples, is there a convenient way to define energy in these scenarios?   Just a system of gravitational waves. A point mass moving in a static (but otherwise arbitrary) space-time. Equivalent (if I'm not mistaken) to a test mass moving in the field of a second much larger mass, the larger mass wouldn't move. Two rotating bodies of similar mass.   Overall, I'm trying to understand what keeps us from associating a potential energy to the metric. When we break the time translation symmetry of a system by introducing an electromagnetic field, we can still conserve energy by defining an electromagnetic potential energy. Why can't we do the same when we break TT symmetry by making space-time curved? When studying systems of linear equations, it's nice to remind people that the last column of the coefficient matrix holds the constants.  This is often done in books by putting a vertical line between the last column and the next to last column.  What is a good way to do this in LaTex? How to fix sudo after "chmod -R 777 /usr/bin"? I entered chmod -R 777 /usr/bin and now sudo is not working.  It says sudo must be setuid root.  Some advice online said to run chown root:root /usr/bin/sudo chmod 4755 /usr/bin/sudo.  On entering chown root:root /usr/bin/sudo it shows an opened in readonly mode error. This is rather difficult to describe with fancy pictures with red circles but it's easily reproducible.  Whenever I click Leave Closed in the re-open queue on MSO the "nominate this question for re-opening" pop-up appears. I normally press escape out of shock and a leave closed vote is recorded.  This is the latest review it's happened to me on:   You can also test it on , I clicked reopen this time and a leave closed vote was recorded but I also got a red popup stating that an error had occurred.  This is also occurring when the Edit &amp; Reopen button is clicked, the edit box opens for a second before being overridden by the popup. 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? I've been reading about a lot of different types of philosophies, religions, ways of thinking etc. I was wondering what are the most recently created philosophical ways of looking at the universe? Where can one find out about new cutting edge ideas?  Example: Before we had Marxism, Communism, Zen, Christianity etc. Now X,Y, and Z have been created building on and evolving the ideas of A,B,C.   Everything I read about are ideas that were created a long time ago. What is new and updated? How can I add a new user as sudoer using the command line? How to set default style for arrow tips in TikZ? How to pass parameters in batch file and zip the file using 7z.exe? I need to create a batch file as per below requirements  1) pass path1 &amp; path2 while running the batch file  exl: testrun.bat E:\ERP\test1.txt E:\ERP\header.txt  2) Need to store these command line paths values in batch file  exl: var1=E:\ERP\test1.txt var2=E:\ERP\header.txt  3) zip the path1 using PKZIPW.exe or 7z.exe and create new o/p zip file  exl: 7z.exe u -tzip E:\Erp\Test1.zip var1  4) copy path2 in newly created o/p file  exl: copy /b var2+test1.zip E:\Erp\Final_output.zip What resampling method is used in the raster 'Save as...' dialog? I received a raster that was resampled using the Save as dialog in QGIS. Now I need to describe the resampling process in a report. But I could not figure out the method used in the resampling. Is it Nearest Neighbor? Bicubic?  This question has been asked before () but the answer does not specify the method used.   I also tried looking at the QGis source code in GitHub, but unfortunately, I could not find where the magic takes place (I'm not a programmer) Let $p(t) =  a_0+\dots +a_nt^n, a_n\neq 0,$ a polynomial with degree $n&gt;0$. Prove that $p:\mathbb{R}\rightarrow \mathbb{R}$ is not bounded, but for any bounded subset $X\subset \mathbb{R}$ the restriction $p\vert X$ is bounded.   I want to prove this in very elementary terms. I mean, without using the notion of limit or even continuity. I can only show this with "strong" results. Is it possible with elementary terms? Limiting double to 3 decimal places Does one work in or on an aeroplane? Testing for linear dependence among the columns of a matrix I have a correlation matrix of security returns whose determinant is zero. (This is a bit surprising since the sample correlation matrix and the corresponding covariance matrix should theoretically be positive definite.)  My hypothesis is that at least one security is linearly dependent on other securities. Is there a function in R that sequentially tests each column a matrix for linear dependence?  For example, one approach would be to build up a correlation matrix one security at a time and calculate the determinant at each step. When the determinant = 0 then stop as you have identified the security who is a linear combination of other securities.  Any other techniques to identify linear dependence in such a matrix are appreciated. Portuguese tab names on profile page A few minutes ago, when I viewed profile pages on my phone, the tabs were all in Portuguese:   On the mobile interface:   On the full site     But it didn't happen on my desktop, and now it's back to normal.  Other users report that the site is normal, but  says it's still happening, so I've decided to post this bug report even though it seems to be working for me at the moment.   Perhaps it's some sort of intermittent problem.  I'm posting this on Meta.SE, since it seemed to happen on both Japanese.SE and ELL.SE, so it doesn't seem to be specific to a particular site. How do I get my Broadcom BCM4313 working correctly? Installing PHP 5.3 in Ubuntu 14.04 I've just installed Ubuntu 14.04 and am trying to install php 5.3. But when I run   apt-get install php5    it's installing php 5.5.9-1ubuntu4. I need to use php 5.3 in the system because the application I'm working on is built with php 5.3.   How can I achieve this? What is the maximum number of pucture repairs an inner tube can take?