let $C =\{c_1,...,c_n\}$ be a prefix-free code and let $l_i =\vert c_i\vert ,L = max\vert c_i\vert$, consider $X$, a random binary string of length $L$,  let $E_i$ be the event that the codeword $c_i$ is a prefix of $X$, what is the value of $\mathbb P(E_i)$, the probability that $E_i$ occurs, use this to provide an alternative proof of McMillans Theorem.  I can calculate $\mathbb P(E_i) = 2^{-l_i}$ for $c_i$ with $l_i&lt;L$, but its all a little messy and unclear how to do this is a clean structured way
Let $\mathcal{C}$ be a finite binary code such that for any two sequences $\{a_n\},\{b_n\}\in \mathcal{C}$ if $a_j\not=b_j$ for some $j$ then the concatenation binary sequence $a_1a_2\cdots a_n\not=b_1b_2\cdots b_n$, the Kraft-Mcmillan Inequality says that $$\sum_{i}\frac{N_i}{2^{i}}\le 1,$$where $N_i$ is the number of codes in $\mathcal{C}$ with length of $i$.  If we assume the code is prefix free, we will have an elegant probabilistic proof at  (consider the infinite binary sequence generated by i.i.d. binomial random variable with probability $1/2$, define $E_x$ be the event that the sequence has prefix $x\in \mathcal{C}$, then $\sum_i\frac{N_i}{2^i}\le \sum_i P[E_x]=  P[\cup_i E_x]=1$). But for the general case one will not necessarily have the events $E_i$s disjoint, I'm curious if similar argument exists for the general case. (The problem is an excercise in the probabilistic method of alon, I guess some probabilistic argument should exist)  (My attempt is that consider the product topology $T=\prod_i\mathcal C$ whose open sets generated by $E_x$s where $x$ is the concatenation codes of $\mathcal{C}$, we define a natural product measure $\mu$ over $T$ by $\mu(E_x)=2^{-|x|}$. Clearly $\mu(T)\le 1$ (as $T$ is a sub-topology of $\prod_i\{0,1\}$), I want to show $\mu(E_x\cap E_y)=0$ for $x\not=y\in \mathcal{C}$, if not we can find $E_z$ such that $E_z\subset E_x\cap E_y$ by compactness, if we can show $z$ can be write as concatenation code start with both $x,y$ we will done, but I get stuck at here.)
How to change permission external hard drive in ubuntu 12.04 I want to copy any file to my external hard drive. But always when I tried to do this I have next output:  cp: cannot create regular file /media/NameOfCard/[file]' : Read only file system  I also tried:  sudo chmod -R 775 /media/NameOfCard sudo chmod -R 777 /media/NameOfCard   But the same effect - ... Read only file system  Update:   After mount I have next output:  nazar_art@nazar-desctop:~$ mount /dev/sda1 on / type ext4 (rw,errors=remount-ro) proc on /proc type proc (rw,noexec,nosuid,nodev) sysfs on /sys type sysfs (rw,noexec,nosuid,nodev) none on /sys/fs/fuse/connections type fusectl (rw) none on /sys/kernel/debug type debugfs (rw) none on /sys/kernel/security type securityfs (rw) udev on /dev type devtmpfs (rw,mode=0755) devpts on /dev/pts type devpts (rw,noexec,nosuid,gid=5,mode=0620) tmpfs on /run type tmpfs (rw,noexec,nosuid,size=10%,mode=0755) none on /run/lock type tmpfs (rw,noexec,nosuid,nodev,size=5242880) none on /run/shm type tmpfs (rw,nosuid,nodev) /dev/sdb1 on /media/sdb1 type vfat (rw) binfmt_misc on /proc/sys/fs/binfmt_misc type binfmt_misc (rw,noexec,nosuid,nodev) gvfs-fuse-daemon on /home/nazar_art/.gvfs type fuse.gvfs-fuse-daemon (rw,nosuid,nodev,user=nazar_art)   Any suggestions?   How to solve this trouble and let permission to copy and delete files to hard drive? Children's book with "square triangles" in parallel universe? What is a 'multi-part identifier' and why can't it be bound? I continually get these errors when I try to update tables based on another table. I end up rewriting the query, change the order of joins, change some groupings and then it eventually works, but I just don't quite get it.  What is a 'multi-part identifier'? When is a 'multi-part identifier' not able to be bound? What is it being bound to anyway? In what cases will this error occur? What are the best ways to prevent it?    The specific error from SQL Server 2005 is:     The multi-part identifier "..." could not be bound.   Here is an example:  UPDATE  [test].[dbo].[CompanyDetail]  SET Mnemonic = [dbBWKMigration].[dbo].[Company].[MNEMONIC],                 [Company Code] = [dbBWKMigration].[dbo].[Company].[COMPANYCODE]  WHERE [Company Name] = **[dbBWKMigration].[dbo].[Company].[COMPANYNAME]**   The actual error:     Msg 4104, Level 16, State 1, Line 3 The multi-part identifier   "dbBWKMigration.dbo.Company.COMPANYNAME" could not be bound. Disk storage graphic and usage amount inconsistent? Send authors an inbox message if their question is closed I think it would really help if you get notified that a question of yours was closed.  Especially for the multi-site power user, usually (at least for me) a question is asked and then left alone for while. I rely on the network-wide inbox to tell me when someone has commented or answered, but if I don't receive any messages then I tend to assume that the question wasn't answered or even noticed. It's only when I check the question a day or two later to see what I can add do I find out it was closed.  This would be especially helpful for duplicates, where the answers you need are already there. A message to the effect of, "Closed as duplicate of question" is just like notifying me that a new answer has been added!  Unfortunately, question closures can remain unnoticed by their askers until months have passed. By then, the questions may either be too old to migrate to the correct site, or may , depending on the specific closing reason. We should notify users of these question closures so that they can improve them and learn from their mistakes.  Could closed questions get added to the inbox? How to do this manually, not installing any tools? Give warning when [super method] is not called On per-site metas, users are given a rank based on meta participation, and this ranking is seen on the default tab of the users page on each per-site meta. How is this participation numerically calculated to give a ranking? Is it by votes? Something else? Are community wiki posts included in this? Increase the visibility of the association bonus by documenting it with the other privileges Show all of my question/answers to me even if they are deleted While I basically understand what a determinant is, I wonder how this idea was developed?  What was the principal idea behind its origination?  I would like to know this so that I can have a better conceptualization of the determinant. pdf of a quotient of uniform random variables Suppose $x_1, x_2$ are IDD random variables uniformly distributed on the interval $(0,1)$.  What is the pdf of the quotient $x_2 / x_1$? How does commands like ls know what its stdout is?  It seems ls is operating different depending on what the target stdout is. For example if I do:  ls /home/matt/tmp    the result is:  a.txt b.txt c.txt   However if I do  ls /home/matt/tmp | cat   the result is (i.e. new line per result):  a.txt b.txt c.txt   The process is passed a file descriptor 1 for stdout right? How does it determine how to format the result? Does the file descriptor reveal information? What is a "double date"? Usage of “when” or “while” in this context He left the house _ I was still sleeping   Would this sentence be better completed with when or with while? Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$. Let $$x_n=\frac{1}{1^2}+\frac{1}{2^2}+\ldots+\frac{1}{n^2},\;\forall n\geq 1.$$ Show that $$\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq \frac{3n-2}{2n-1}.$$   I tried to prove it by induction  but I got a contradiction. Hide trivial comments Expected Value of Max of IID Variables What is the expected value of the maximum of 500 IID random variables   with uniform distribution between 0 and 1?   I'm not quite sure of the technique to go about solving something like this. Could anyone point me the right direction?  Thanks I am getting an 'access is denied' error when I attempt to delete a folder that is not empty. I used the following command in my attempt: os.remove("/folder_name").   What is the most effective way of removing/deleting a folder/directory that is not empty? Merge around a face So, I'm working on a control station that has buttons. It is a one-mesh creation, that means, I don't want to create buttons and things like that separated from the main mesh, but as you can see, I have problems. I would like to make the highlighted parts to be one with the main part around them. I've tried everything, and I have this problem on multiple creations. Any idea?  Update: I've added another example of the problem. I want them to be one face, instead of two. (2nd picture)