I know how to prove real number is uncountable. But since both real number and natural number are linear ordered, it feels like there is some one to one relation between those two. Like the infinite hotel story, since they are both ordered, we can move them in the hotel one by one. But this is obviously wrong, I just can not figure out why. Thanks.
Question 1: Why are irrational numbers uncountable and rationals countable?  I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers there will always be a irrational number. But now it has come to my attention that the same holds for rational numbers. So I can't really see the difference between them. What is it that makes the difference? Why can we line up all rational numbers one-to-one with the natural numbers, but not the irrationals with the naturals?  Sorry for a perhaps basic question, but I really, really struggle with it. Thanks.
Which are the spec bits in this cache? I have a direct mapped cache of size S with the line size L. The cache is physically indexed and tagged. The physical address is 50 bits, numbered from 0 to 49 (with 0 being the least significant bit). The machine has a word size of 4 bits and the memory is byte-addressable. Which bits of the 50 bits are compared with the address tag? How can I correct a reviewer's opinion? Is it possible to get old suggestion system back in Firefox 43.0? With the new update, the first suggested link is just domain. In older versions, the first suggested link was the page I visited the most and I got used to this great feature.  Is it possible to remove the new domain suggestion, for example "Visit ebay.co.uk"?  See image attached Does my enemy get to make a saving throw for half damage for absolutely any spell I cast? So me and a group of friends recently started to play D&amp;D. I became a sorcerer. My question is: can the enemy I am attacking, no matter what spell I use, make a saving throw for half damage or does that only apply for certain spells?  For example, when I use my cantrip fire bolt, does the targeted opponent get to roll a saving throw for half damage against the spell, even though the description doesn't say anything about saving throws? I am going through Shankar's Principles of Quantum Mechanics and am having trouble finding the free particle propagator $U(t)$ that satisfies $$\lvert\psi (t)\rangle = U(t)\lvert\psi (0)\rangle$$ due to the degeneracy of the $E$ eigenkets. Shankar says that if the Hamiltonian has degenerate eigenvalues, we change the propagator equation from $$U(t) = \sum_{E} \lvert E\rangle\langle E\rvert e^{-iEt/{\hbar}}\tag{4.3.13}$$ to $$U(t) = \sum_{E} \sum_\alpha \lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar} $$ where $\lvert E, \alpha \rangle $ are the orthonormal eigenkets for the $E$ eigenspace. This makes sense to me, because it seems like whenever we deal with degeneracy to use a projection. He also says for a Hamiltonian with no degeneracy to change the sum into an integral, so I imagine that for a Hamiltonian with a continuous, degenerate spectrum we would have this propagator: $$ U(t) = \sum_\alpha\int_\Re dE\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}$$  I tried applying this to the free particle and had some trouble. In the book, after solving for the eigenkets and eigenvalues, $\lvert E\rangle = \lvert p\rangle$ and $p = \pm\sqrt{2mE}$, he chooses the propagator as a function of the momentum eigenvalues $p$ so that $$U(t) = \int_\Re dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m).\tag{5.1.9}$$ I understand how he got to this, but I'm having some trouble because this doesn't appear to be the same expression that I have above. Splitting the integral up, we have $$U(t) = \int_{-\infty}^0dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ The left integral can changed to an integral over $-p$, and the bounds switch from $0$ to $\infty$ such that: $$U(t) =  \int_{0}^\infty d(-p)\lvert -p\rangle\langle -p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ Now we can substitute and interchange $\lvert p\rangle = \lvert E, + \rangle$,  $\lvert -p\rangle = \lvert E, - \rangle$, $E = p^2/2m$, and $dp = \pm mdE/\sqrt{2mE}$ to get: $$U(t) = \sum_{\alpha = \pm} \int_0^\infty dEm/\sqrt{2mE}\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}. $$ This is the result that we are asked to prove in Exercise 5.1.1. Why is this not the same as the first expression for the propagator? Where did the extra $ m/\sqrt{2mE} $ come from? Am I just wrong in assuming that that is the general expression for a Hamiltonian with continuous and degenerate eigenvalues? Format a date in XML via XSLT How to create a GUID / UUID I'm trying to create globally-unique identifiers in JavaScript.  I'm not sure what routines are available on all browsers, how &quot;random&quot; and seeded the built-in random number generator is, etc. The GUID / UUID should be at least 32 characters and should stay in the ASCII range to avoid trouble when passing them around. if $a,b,c$ are non zero positive reals prove $$\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$$ I have used A.M  G.M  inequality as follows:  $$a^3+b^3+c^3 \ge 3abc$$ adding $abc$ both sides we get  $$a^3+b^3+abc \ge 4abc-c^3=c(4ab-c^2)$$ so  $$\frac{1}{a^3+b^3+abc} \le \frac{1}{c(4ab-c^2)}$$ But since $a,b,c$ are positive reals  $$4abc-c^3 &lt; 4abc$$ so  $$\frac{1}{4abc-c^3} \gt \frac{1}{4abc}$$  but i am unable to proceed here "To work for" vs. "to work in" vs. "to work at" How to let PHP to create subdomain automatically for each user? How do I create subdomain like  ? Do i have to access htaccess somehow? Is it actually simply possible to create it via pure php code or I need to use some external script-server side language?  To those who answered: Well, then, should i ask my hosting if they provide some sort of DNS access?? Why is it considered so dangerous to travel in South Africa? As, the above link shows, other countries have similar murder rates but they are not in the radar of international travelers.  For example, as seen in the list, Puerto Rico has similar statistics. But, personally, I never heard of Puerto Rico to be a dangerous place to travel.  Why is it considered so dangerous to travel in South Africa? Is that due to the incompetent law enforcement agencies? Do redirection operators always open file descriptors in parallel? 1. Consider snippet#1:  $ cat test.txt &gt; test.txt cat: test.txt: input file is output file   It seems that cat makes its input file descriptor point to test.txt and then when it tries to set its output file descriptor to test.txt it throws the above error. Here it seems that cat is aware of the redirect operator and so proceeds to attempt to set the output file descriptor to test.txt  2. Consider snippet#2:  $ cat 1.txt 1:CAT 2:dog $ sed 's/cat/CAT/g' test.txt 1:CAT 2:dog $ sed 's/cat/CAT/g' test.txt &gt; test.txt $ cat test.txt # Note that test.txt is now empty $   Here we see that sed opens test.txt (last argument) in read mode and at the same time sets test.txt as its file output descriptor. Also the '&gt;' operator overwrites the contents of the file BEFORE sed starts to read from it.   I am aware that commands in a pipeline execute in parallel but have not come across any info on how redirection operators behave. Any supporting links would be helpful. Replace non-numeric with empty string Quick add on requirement in our project.  A field in our DB to hold a phone number is set to only allow 10 characters.  So, if I get passed "(913)-444-5555" or anything else, is there a quick way to run a string through some kind of special replace function that I can pass it a set of characters to allow?  Regex? In .NET, what's the best way to prevent multiple instances of an app from running at the same time? And if there's no "best" technique, what are some of the caveats to consider with each solution? How can we run a command stored in a variable? $ ls -l /tmp/test/my\ dir/ total 0   I was wondering why the following ways to run the above command fail or succeed?  $ abc='ls -l "/tmp/test/my dir"'  $ $abc ls: cannot access '"/tmp/test/my': No such file or directory ls: cannot access 'dir"': No such file or directory  $ "$abc" bash: ls -l "/tmp/test/my dir": No such file or directory  $ bash -c $abc 'my dir'  $ bash -c "$abc" total 0  $ eval $abc total 0  $ eval "$abc" total 0 Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$. Why is $H(k\mathbin\Vert x)$ not a secure MAC construction? If $H(m)$ is a secure hash function, can't we implement a MAC using $H(k\mathbin\Vert m)$?  However, it seems the more widely used MACs, such as NMAC and  (both originally defined in ) use a much more complicated scheme. Why is this concatenation scheme insecure? Sort and count number of occurrence of lines I have Apache logfile, access.log, how to count number of line occurrence in that file? for example the result of cut -f 7 -d ' ' | cut -d '?' -f 1 | tr '[:upper:]' '[:lower:]' is  a.php b.php a.php c.php d.php b.php a.php   the result that I want is:  3 a.php 2 b.php 1 d.php # order doesn't matter 1 c.php I am wondering how to show that $\mathbb{Z}[\sqrt {10}]$ is not a UFD.    My only idea is to show that there are two factorizations of $10$, say, $ab, uv$ such that $a$ is not a unit times $u$ or $v$. In this ring $10=2\cdot5=\sqrt {10}\cdot \sqrt {10}$, so it suffices to show $2$ is not a unit times $\sqrt {10}$. Suppose $2=\sqrt {10}(a+b\sqrt{10})=a\sqrt{10}+10b$. Then $a=0$ since $\sqrt{10}$ is not rational. So $10b=2$, which has no integer solutions. So $\mathbb{Z}[\sqrt {10}]$ is not a UFD.  Is my reasoning correct? What are the flaws? How can I install the most used plugins like flash player and mp3 in Ubuntu?