how to prove that $\#\mathcal{P}(\mathbb{N}) = \#\mathbb{R}$? I'm thinking in associate a subset of $\mathbb{N}$ in a real number by the decimal representation, but i'm failed.
I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it?
threeparttablex table notes for tabu tabulars with less than \textwidth Explain what is a transistor in plain English! My internet connection is very slow. I suspect someone in the network is using a BitTorrent client which takes huge amount of bandwidth.  Is there a way to detect who is using my bandwidth (network sniffing tools etc)?  I should have pointed out that I have no access to the router, what I am looking for is a way of detecting someone's MAC/IP if she uses Bittorrent. Why are the characters on the keyboard all capitalized? Why are all the letters (A, B etc) and all the writings (Home, Page Up etc) on the keyboard, capitalized?    Especially, the letters, because a new user could rightly expect uppercase letters to be entered upon pressing those keys to find the contrary. #!/bin/bash - no such file or directory I've created a bash script but when I try to execute it, I get   #!/bin/bash no such file or directory   I need to run the command: bash script.sh for it to work.  How can I fix this? Why don't we consider jerk in physics classes? When I got more into physics, I started asking myself if just like acceleration represents the growth of speed, something else could also represent the growth of acceleration itself. And it came that is exists and is called jerk. Before I thought about this, I thought that acceleration was the highest derivative that made sense on physics.  My question is, why can we perform accurate calculations without account for jerk? I never needed it in a single physics class! I've also read that it is seldom used.  Also, while position, speed and acceleration seem familiar to me due to everyday experience, why doesn't jerk seem familiar to me? I never tell someone when driving a car "you're jerking too much", whereas I could say "you are accelerating too much".  Is it that jerk has little influence because it is almost constant? Even if it is not, what about higher order derivatives? Why don't derivatives higher than acceleration have much influence on calculations? We perform a lot of reasoning in everyday life without resorting to it! At what point is the ~/.bashrc file created? What is the correct way to write 1.5 hours? I'm a bit confused in describing 1.5hrs in words.   Is writing one and a half an hour correct or should it be one and half hour?  E.g:     I'll see you there in one and a half an hour.     OR       I'll see you there in one and half hour.   Or is there any other correct way of writing this? I want to create a text file which will add each command I am going to run on the terminal as a new line entry in the text file. i.e., I need to keep a record of what are the things I'm doing in the terminal. Further if I can keep a record of outputs I'm getting in the terminal, along with the inputs in the terminal, as a text file, it will be much better.   Each time I forget a command I have to search again in the Internet for the command.  Can I can make a file like above? There's a web services I want to call in my application, I can use it with importing the WSDL or by just use "HTTP GET" with the URL and parameters, so I prefer the later because it's simple thing.  I know I can use indy idhttp.get, to do the job, but this is very simple thing and I don't want to add complex indy code to my application.  UPDATE: sorry if I was not clear, I meant by "not to add complex indy code", that I don't want add indy components for just this simple task, and prefer more lighter way for that. I've recently started maintaining someone else's JavaScript code. I'm fixing bugs, adding features and also trying to tidy up the code and make it more consistent.  The previous developer used two ways of declaring functions and I can't work out if there is a reason behind it or not.  The two ways are:  var functionOne = function() {     // Some code };     function functionTwo() {     // Some code }   What are the reasons for using these two different methods and what are the pros and cons of each? Is there anything that can be done with one method that can't be done with the other? Why does locking the rear tires on a vehicle cause it to spin? are often used in motor-sports and stunt driving to turn around tight corners.  Essentially, the driver applies the handbrake as he enters the turn, in order to "kick out" the back end and slide the vehicle around the corner.  However, applying the handbrake hard enough to lock the rear wheels while traveling straight will cause the rear of the vehicle to "kick out".  Given enough speed, the vehicle will spin 180 degrees.  Why do sliding wheels on a car want to "get in front" of rolling wheels? If I utilize the two columns of multicols for my Kurdish sequence of numbers, it arranges the items in column like the following figure:   But I want to arrange the items in rows splitted by some columns (two columns in this case) alike the following figure:   And for the case of "three columns", the output would be alike the following figure:   How may I achieve the functionality? And I don't want to use the sequence of the following commands:  \begin{multicols}{2}‎ \begin{enumerate}[$1)$,noitemsep] \item ‎ Yek \item ‎ Du \end{enumerate} \end{multicols}‎   Instead, I want to specify the split count of the columns and then have the items arranged in the desired arrangement automatically. For the record, I've check  post, but it's too generalized. If a derivative of a continuous function has a limit, must it agree with that limit? Evaluate the integral $\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$ This integral looks a lot like an elliptic integral, but with cubes instead of squares: $$I(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$$ Let's consider $a,b&gt;0$ for now. $$I(a,a)=\int_0^\infty \frac{dx}{x^3+a^3}=\frac{2 \pi}{3 \sqrt{3} a^2}$$ I obtained the general series solution the following way. Choose $a,b$ such that $a \geq b$, then: $$I(a,b)=\frac{1}{a^2} \int_0^\infty \frac{dt}{\sqrt{(t^3+1)(t^3+b^3/a^3)}}=\frac{1}{a^2} I \left(1, \frac{b}{a} \right)$$ $$\frac{b^3}{a^3}=p, \qquad I \left(1, \frac{b}{a} \right)=I_1(p)$$ $$I_1(p)=\int_0^\infty\frac{dt}{\sqrt{(t^3+1)(t^3+p)}}=2 \frac{d}{dp} J(p)$$ $$J(p)=\int_0^\infty\sqrt{\frac{t^3+p}{t^3+1}}dt=\int_0^\infty\sqrt{1+\frac{p-1}{t^3+1}}dt=$$ $$|p-1| \leq 1$$ $$=\sum_{k=0}^\infty \left( \begin{array}( 1/2 \\ ~k \end{array} \right) (p-1)^k \int_0^\infty \frac{dt}{(t^3+1)^k}$$ Now this is the most problematic part. The first integral of this series diverges. However, it's a constant in $p$, so if we differentiate, it formally disappears: $$I_1(p)=2 \sum_{k=1}^\infty \left( \begin{array}( 1/2 \\ ~k \end{array} \right) k (p-1)^{k-1} \int_0^\infty \frac{dt}{(t^3+1)^k}$$ Now, every integral in this series converges. The integtals can be computed using the Beta function, if we substitute: $$t^3=\frac{1}{u}-1$$ Finally, we rewrite: $$I_1(p)=\frac{\Gamma (1/3)}{3 \sqrt{\pi}} \sum_{k=1}^\infty \frac{k^2}{k!^2} \Gamma \left(k- \frac{1}{2}\right) \Gamma \left(k- \frac{1}{3}\right) (1-p)^{k-1}$$ Or, using the Pochhammer symbol:  $$I_1(p)=\frac{2 \pi}{3 \sqrt{3}} \sum_{k=0}^\infty \frac{(k+1)^2}{(k+1)!^2} \left(\frac{1}{2}\right)_k \left(\frac{2}{3}\right)_k (1-p)^k$$  My questions are:  Is the method I used valid (see the 'problematic part')? How to get this series into a Hypergeometric function form? Is there any 'arithmetic-geometric mean'-like transformation (Landen's transformation) for this integral? How to go about finding it?  If the method I used is correct, it can be used for any integral of the form ($m \geq 2$): $$I_m(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^m+a^m)(x^m+b^m)}}$$ How do I connect to a WPA wifi network using the command line? So basically, I'd like to use iwconfig to connect to my wifi network when I'm not inside X. But I just don't see a way to do it in the man page. Any idea? Calculate posterior distribution (gamma-prior, poisson-likelihood) I want to calculate the posterior distribution given a gamma-prior and a poisson likelihood. The task is from a textbook and I just have the solutions (without a walkthrough). Please find all given information in the following:  Given information:  $m$ countries each with a population of $n_i$; $y_i$ is the number of people having a disease; $y_i$ is a poisson random variable with mean $\frac{n_i*\lambda}{100}$  The prior distribution for $\lambda$: $\lambda \sim Gamma(a_0,b_0)$  Task:  Show that the posterior distribution is also a gamma distribution with $a_m = a_0 + \sum_{i = 1}^{m}y_i$ and $b_m = b_0 + \sum_{i = 1}^{m} \frac{n_i}{100}$  My solution:  $Posterior \propto Likelihood \times Prior$  $p(\theta|y) \propto \prod_{i=1}^{m}(\theta^{y_i}e^{-\theta})\times \theta^{a_0-1}e^{-b_0\theta}$  $p(\theta|y) \propto \theta^{\sum_{i = 1}^{m}y_i}e^{-m\theta}\times \theta^{a_0-1}e^{-b_0\theta}$  $p(\theta|y) \propto \theta^{\sum_{i = 1}^{m}y_i+a_0-1}e^{-\theta(m+b_0)}$  $a_m = a_0 + \sum_{i = 1}^{m}y_i$  $b_m = b_0 + m \neq b_0 + \sum_{i = 1}^{m} \frac{n_i}{100}$  I think I am doing a mistake, because I don't know how to incorporate the information that the mean of $y_i$ is given as $\frac{n_i*\lambda}{100}$. Or do I just have to transform $b_m$ somehow? Can someone may give me a hint?   Thank you already in advance!  Edit 1: Considering first advices  $Likelihood: p(y | \theta) \propto \prod_{i=1}^{m}(\frac{n_i\lambda}{100})^{y_i}e^{-\frac{n_i\lambda}{100}}$  $\propto \prod_{i = 1}^{m} ((\frac{n_i}{100})^{y_i})\lambda^{\sum_{i=1}^{m}y_i}e^{-\lambda\sum_{i=1}^{m}\frac{n_i}{100}}$  $\propto \lambda^{\sum_{i=1}^{m}y_i}e^{-\lambda\sum_{i=1}^{m}\frac{n_i}{100}}$  Is the transformation of $\prod_{i=1}^{m}$ I did from the first to the second step correct? When I do upgrade it gets stuck here:  user@plato:~# sudo apt-get upgrade Reading package lists... Done Building dependency tree        Reading state information... Done Calculating upgrade... Done The following packages will be upgraded:   bind9-host coreutils cpio curl dnsutils libbind9-90 libc-bin libc-dev-bin   libc6 libc6-dev libcgmanager0 libcurl3 libcurl3-gnutls libdns100   libdrm-intel1 libdrm-nouveau2 libdrm-radeon1 libdrm2 libisc95 libisccc90   libisccfg90 libjasper1 liblwres90 libplymouth2 libsepol1 libssl-dev   libssl-doc libssl1.0.0 linux-libc-dev mime-support multiarch-support ntp   ntpdate openssl plymouth plymouth-theme-ubuntu-text python3-distupgrade   ubuntu-release-upgrader-core unzip 39 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 16.0 MB of archives. After this operation, 12.3 kB of additional disk space will be used. Do you want to continue? [Y/n] y 0% [Connecting to us.archive.ubuntu.com (2001:67c:1562::15)] ░   When I do update it gets stuck here:  user@plato:~# apt-get update 0% [Connecting to us.archive.ubuntu.com (2001:67c:1562::14)] [Connecting to sec░   I'm able to ping us.archive.ubuntu.com just fine:  user@plato:~# ping us.archive.ubuntu.com PING us.archive.ubuntu.com (91.189.91.23) 56(84) bytes of data. 64 bytes from economy.canonical.com (91.189.91.23): icmp_seq=1 ttl=54 time=81.5 ms 64 bytes from economy.canonical.com (91.189.91.23): icmp_seq=2 ttl=54 time=81.5 ms 64 bytes from economy.canonical.com (91.189.91.23): icmp_seq=3 ttl=54 time=81.5 ms 64 bytes from economy.canonical.com (91.189.91.23): icmp_seq=4 ttl=54 time=81.4 ms ^C --- us.archive.ubuntu.com ping statistics --- 4 packets transmitted, 4 received, 0% packet loss, time 3004ms rtt min/avg/max/mdev = 81.425/81.528/81.582/0.355 ms   What can i do to fix this?    What I have tried so far.  I've tried following directions on this blog:   To force apt-get to use IPv4 exclusively   Create a file in /etc/apt/apt.conf.d/99force-ipv4 Add this line to newly created file Acquire::ForceIPv4 false; Apt-get will now use IPv4 exclusively What can I do if my USB flash drive is write-protected or read-only? When I plug in my USB flash drive, it shows up on my computer as write-protected or read-only. I am unable to transfer data to it, nor can I modify or delete any files already stored on it. I also cannot repartition or reformat the drive using Windows Disk Management, DiskPart, GParted, or other tools. The drive does not have a write-protect switch.  Why did this happen and what can I do about it? Is there a way to remove the write protection?  (Note that this can happen with some memory cards, too, as they often use controllers similar to those used in flash drives. In some cases, the system may report that the drive or card was formatted successfully even though it was never actually formatted; the original data reappears when the device is reinserted.)     This question comes up often and the answers are usually the same. This post is meant to provide a definitive, canonical answer for this problem. Feel free to edit the answer to add additional details. country code TLD vs second level ccTLD I'm really worried about professionalism.  My site would be something like http://mywebsite.uy instead of http://mywebsite.com.uy.  What do you think about this practice? What should I have in mind at the moment of choosing one?