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A generalized version of the second Borel-Cantelli lemma says     Theorem 5.3.2. Second Borel-Cantelli lemma, II. Let $\mathcal F_n, n \ge 0$ be a filtration   with $F_0 = \{\emptyset, \Omega\}$ and $A_n , n \ge 1$ a sequence of events with $A_n ∈ \mathcal F_n$ . Then   $$ \{A_n \,i.o.\} = \left\{\sum_{n \ge 1} P (A_n |\mathcal F_{n−1}) =\infty \right\}. $$   An exercise of this lemma is     Exercise 5.3.6. Show    $\sum_{n \ge 2} \mathbb P (A_n | \cap_{m=1}^{n−1} A_m^c ) = \infty$ implies $P (∩_{m \ge 1} A_m^c ) = 0$.   I think we can actually prove that $\cap_{m \ge 1} A_m^c = \emptyset$. My proof is like this: Let $B_n = \cup_{m = 1}^n A_n$. Let $\mathcal F_n = \sigma(A_1,\dots,A_n)$, which forms a filtration. It is easy to verify that $$ \mathbb P (A_n | B_{n-1}^c ) = \mathbb P (A_n | \mathcal F_{n-1})(\omega) $$ when $\omega \in B_{n-1}^c$.  Assume there exist $\omega \in \cap_{m \ge 1} A_m^c$. It follows from the previous lemma that $$ \omega \in \left\{\sum_{n \ge 1} P (A_n |\mathcal F_{n−1}) =\infty \right\} = \{A_n \,i.o.\}. $$ In other words, $\omega \in \cap_{m \ge 1} \cup_{n \ge m} A_n$, which contradicts $\omega \in \cap_{m \ge 1} A_m^c$.  Is this the right proof? We will generate a csv file with below values  yp1234,577,1,3 yp5678,577,3,5 yp9012,132,8,9   I need to extract data and create files based on second column. If it's 577 then the whole line has to be extracted and placed in a separate file. I mean I need a file having lines with second column as 577 alone and another file with second column as 132 alone  I tried using IF but didn't work When I have equations, as in  \begin{equation} ... \end{equation}   Can I arbitrarily assign the numbers to the equations? Instead of the usual order (1), (2), (3), ..., I'd like something like (1), (2.1), (2.2), (2.3), (3.1), (3.2), (4), ... I know that there's a difference between relativistic and rest mass. Relativistic mass is "acquired" when an object is moving at speeds comparable to the speed of light. Rest mass is the inherent mass that something has regardless of the speed its moving at.  When fusion happens, a certain percentage of the rest mass of two hydrogen atoms is converted into energy. I understand that $E = m_{rest}c^2$.   How does the rest mass of an object turn into energy? Does this involve particle physics? Doesn't this suggest that rest mass can be viewed as some sort of potential energy? It seems to me that it's a sort of locked energy that has to undergo a certain process to expel its energy to surrounding particles.  Also, Why does only a small percentage of rest mass turn into energy? Why not all of it. What dictates how much rest mass gets converted into energy? Is the set of real matrices diagonalizable in $\mathcal M_n(C)$ dense in the set of all companion matrices? Let $\mathcal E=\{A \in \mathcal M_n(\mathbb R): A \text{ is of companion form} \}$. That is, $A$ has the following form \begin{align*} A=\begin{pmatrix}  0 &amp; \cdots &amp; 0&amp; -a_{0} \\  1 &amp; \cdots &amp; 0 &amp; -a_{1}\\ \vdots &amp;\ddots &amp; \vdots &amp;\vdots \\ 0 &amp;\cdots &amp; 1 &amp; -a_{n-1} \end{pmatrix}. \end{align*} Let $\mathcal D = \{A \in \mathcal M_n(\mathbb R): A \text{ is diagonalizable in $\mathcal M_n(\mathbb C)$}\}$. We know $\mathcal D$ is dense in $\mathcal M_n(\mathbb R)$. I am wondering whether $\mathcal E \cap \mathcal D$ is dense in $\mathcal E$. The set $\mathcal E$ is closed. It is not clear to me whether this statement is true. How do I go about porting Ubuntu Touch to different devices? How do I spamtrap with Postfix? How do I setup spamtraps with Postfix so clients that send to trapped addresses are prevented from sending me more spam? Apache 2.4 - 403 HTTP status code WPF DataGrid : CanContentScroll property causing odd behavior Is there an easy method for installing binary builds of glibc? Time and time again I see questions such as these:       And these are the types of solutions we're typically pushing:        Is this really the best we can do? Aren't there binary builds of GLIBC that we can simply unzip into a directory such as /opt/myglibc, and set the $LD_LIBRARY_PATH or whatever and run whatever application we want, without issues?   An application, such as the newer builds of Chrome (28+) that appear to require GLIBC 2.14?  NOTE: This thread titled:  on tecmint.com is what ultimately got me thinking about this.  References How can I add a Login link to the "Main navigation" menu?  When a user clicks on the link, it should take them to the /user/login page. Once they login, the link should change to say "Logout" instead of "Login".  Is there any way to add such a link? Is there a way to transfer thousand of small files faster on Windows 7? I noticed that if I transfer a few large files between two hard drives, it's pretty speedy, at around at least 30&nbsp;MB per second, but if I transfer thousands of files less than 5&nbsp;KB, it is pretty damn too slow.. around 1 to 2&nbsp;MB per second.   Is there a way to speed up the copy/paste process with thousands of small files on Windows 7? Does a family of a distribution have a different definition for statistics than in other disciplines?  In general, a  is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is varied. Such families are used, for example, to .  For statistics, a family according to  is the result of varying the shape parameter. How then can we understand that the gamma distribution has a shape and scale parameter and only the generalized gamma distribution has, in addition, a location parameter? Does that make the family the result of varying the location parameter? According to @whuber the meaning of a family is implicitly A "parameterization" of a family is a continuous map from a subset of ℝ$^n$, with its usual topology, into the space of distributions, whose image is that family.  What, in simple language, is a family for statistical distributions?  A question about relations among of the statistical properties of distributions from the same family has already generated considerable controversy for a  so it seems worthwhile to explore the meaning.  That this is not necessarily a simple question is born out by its use in the phrase , which has nothing to do with a family of curves, but is related to changing the form of the PDF of a distribution by reparameterization not only of parameters, but also substitution of functions of independent random variables. How to allow Page Up in GNU Screen? I use Vim to edit text files on Linux. Commonly I connect to remote Linux machines using PuTTY from Windows.  I use GNU Screen commonly as well.  When using Vim inside Screen I am unable to use the Page Up key because Screen interprets it as a command to start copy mode.    I would like to use Page Up for its regular up-paging functionality.  How can I configure Screen to do this? How does A* pathfinding work? At which moment does the 'Surprised' state disappear? I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$.  I already know the logical Proof:  $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$  Hence summation can be expressed as:  $$\binom{n}{0}\binom{n}{n} + \binom{n}{1}\binom{n}{n-1} + \cdots + \binom{n}{n}\binom{n}{0}$$  One can think of it as choosing $n$ people from a group of $2n$  (imagine dividing a group of $2n$ into $2$ groups of $n$ people each. I can get $k$ people from group $1$ and another $n-k$ people from group $2$. We do this from $k = 0$ to $n$. Pipe to multiple files in the shell I have an application which will produce a large amount of data which I do not wish to store onto the disk. The application mostly outputs data which I do not wish to use, but a set of useful information that must be split into separate files. For example, given the following output:  JUNK JUNK JUNK JUNK A 1 JUNK B 5 C 1 JUNK   I could run the application three times like so:  ./app | grep A &gt; A.out ./app | grep B &gt; B.out ./app | grep C &gt; C.out   This would get me what I want, but it would take too long. I also don't want to dump all the outputs to a single file and parse through that.  Is there any way to combine the three operations shown above in such a way that I only need to run the application once and still get three separate output files? Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$ Can this integral be solved with contour integral or by some application of residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$  It has two poles at $\pm i$ and  branch point of $-1$ while the integral is to be evaluated from $0\to \infty$. How to get $\text{Catalan Constant}$? Please give some hints. How many roots have a complex number with irrational exponent? If a rational exponent on a complex number $z^q$ is the representation of a finite number of roots, then if the exponent is irrational this mean that there is infinite countable roots?  If this is the case... the cardinality of the number of roots is the same for any irrational exponent? Thanks in advance.  NOTE: there are different questions about irrational exponents but no one answer what Im searching so please dont mark this as repeated.