Is $2^{|\mathbb{N}|} = |\mathbb{R}|$? If so, how?  I was reading the Wiki page on the , and it says "Moreover, $\mathbb{R}$ has the same number of elements as the power set of $\mathbb{N}$", but I don't see how this is true?  I feel like it has something to do with binary, but I'm not too sure how it works? Do I have to show a map of all reals can be done in binary? I'm just very confused, and any advice would be appreciated!
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?
I know they have their adherents, but do more or less esoteric branches of mathematics such as Category Theory and/or Quantum Logic provide powerful tools for new theory development or are they just occasionally-useful presentational frameworks?  More particularly, is it worth investing the time to study these formalisms? Connection pooling in PHP Disable chat migration notification if one of the users has insufficient rep? How can I make Firefox open magnet-links in Transmission? I can't find out how to make Firefox open magnet-links in Transmission.   There is no option to make it the standard program for magnet-links in its properties, and as I am not experienced with the filesystem so I can't add it as the standard program in Firefox either. How to pass command output as multiple arguments to another command I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that? The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of that? "Add" button doesn't work with JavaScript off I'm using stackoverflow.com with JavaScript off. I accept this comes with somewhat less functionality, however for the most part SO works fine.  If I look at the home page (), every single button on there except for two types works with JS off.  The first second is that the "x Votes; y answers;z views" mouse-over indicates they're links but clicking them does nothing. This one is less of an issue because the user can still click the question title, and they're only implied buttons anyway.  The second (and bigger) issue is the "Add" buttons next to "Favorite Tags" and "Ignored Tags". These are explicit buttons that don't do anything with JS off. There is no working JS substitute for them (the "tag subscriptions" link itself works, but the page it goes to doesn't work with JS off, but that's another bug report). Number of solutions of an arithmetic function's equation Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation}  where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of distinct prime factors of $x$.  I would like to know if this equation holds for finitely many values of $x$ or infinitely many.  If one uses asymptotic formula (from number theory) it can be written that \begin{equation} \omega(x) \sim \ln x / \ln \ln(x) \end{equation} (from )  and  \begin{equation} \pi(x) \sim x/\ln(x), \end{equation} so the above equation can be rewritten (after simplification) as \begin{equation} 2x\ln(2)=\ln(x)\ln\ln(x)\ln(2x) +f(x) \end{equation}  where $f(x)$ is an error term.  If  I plot left side (which is linear, supposed to be a straight line) and right side(a curve),assuming $ f(x) $ is 100% accurate, then the equation implies(?!) that the right will intersect left side. Since line can intersect a curve ($ln  $ function) finite times, so there are finite intersection point. So, can I say that there are only finitely many values of $x$?  graph example-   here f(x) is assumed as x + sinx  without the f(x)- How can one show there is no holomorphic function $f$ on the open unit disc $\mathbb{D}$ such that it extends continuously to $\frac{1}{z}$ on $\partial\mathbb{D}$? I mean $f$ takes value $\frac{1}{z}$ when $|z|=1$ and continuous here.  Thank you. I would like to know the purpose of nohup these days. I ask because I read this  Nohup is short for “No Hangups.” It’s not a command that you run by itself. Nohup is a supplemental command that tells the Linux system not to stop another command once it has started. That means it’ll keep running until it’s done, even if the user that started it logs out. The syntax for nohup is simple and looks something like this: nohup sh your-script.sh &amp; Notice the “&amp;” at the end of the command. That moves the command to the background, freeing up the terminal that you’re working in. Nohup works with just about any command that you run in the terminal. It can be run with custom scripts as well as standard system commands and command line utilities.  With using linux distributions such as SLES 11.4, or RHEL/CentOS 7.x, I can  remote log in to linux over the network via SSH run the simple program below, or any other, doing ./a.out &amp; log out of linux; type exit in the putty SSH terminal my sample program below, or any other including bash, csh, or tcsh shell scripts, run to completion just fine without the need for nohup typing exit in putty to close an SSH session, I would think that qualifies as a logout?  Is there any purpose or value with nouhup today?  Is it a kernel &gt;= 3.x thing? For however long I can remember (kernel &gt;= 2.6) I've never used nohup and have always just use &amp; with no problems. Can someone give me a practical scenario where nohup would be used? #include &lt;stdlib.h&gt; #include &lt;stdio.h&gt;  /*     sample C code, to print numbers to a file named zz.tmp     the numbers 0 to 29 over 30 seconds */  int main ( int argc, char *argv[] ) {    FILE *fp;    int i;     i = 0;    fp = fopen(&quot;zz.tmp&quot;, &quot;w&quot; );     while ( i &lt; 30 )    {       fprintf( fp, &quot;%d\n&quot;, i++ );       sleep( 1 );    }    fclose( fp );    return 0; }  mysleep.sh #!/bin/bash  sleep 1000  On Centos 7.7, optiplex pc login with keyboard &amp; mouse, my account is bash. I do nohup ./mysleep.sh and then control-z.  When typing jobs I see [1]+ Stopped nohup ./mysleep.sh A ps -ef | grep mysleep shows that process id existing, but if I close the terminal window via X in upper right corner, that process goes away so it seems nohup is not working? Word for "willing to try new and unfamiliar things" Example:  A: Hey, do you like to do bungee jumping? B: Um, OK. Haven't done it before, but let's try it.  I'm not looking for adventurous or something similar. That word should also fit in less action-packed activities:  A: I have a new board game. Do you like to play? B: I don't know this game and, actually, I don't like these varieties of games but I'll play. Maybe it's fun.  First I thought about open-minded but I think this means more willing to listen to people, think about their ideas, arguments, ... and accept them (not necessarily agree). What word describes person who is willing to try activities he doesn't know or hasn't experienced before? Alternative definition of hyperbolic cosine without relying on exponential function Looking for a basic PostGIS viewer to use as basis for own application I'm looking for a rather basic PostGIS viewer. It should be possible to pan, zoom and retrieve coordinate information on mouse click. No need to support coordinate transformations. It should introduce as little further dependencies as possible. Preferably written in C++. Maximum length of a USB Cable? Can I send in for repair a lens that has a shattered front element?     UPDATE: The front element was not shattered, it was the UV filter. Factory diagnosis: Lens assembly had been impacted; cleaned and checked.  Helpful notes by Crowley, David Richerby (x2), Michael Clark (x2), and   mattdm (on element price vs UV filter).      Canon cleaned and checked my lens back to like new quality for $179 labor, no parts, not including shipping.      Jim Garrison answered my question, but Crazy Dino has the best info all around for a lens element shatter.   Original Question:  I recently opened my camera bag to find a shattered front element on a Canon EF 16-35mm f/2.8 L II USM lens.   There are a lot of micro glass shards on the second element.  Does anyone have experience with this sort of thing?  Some of the following are questions I have:   Can I get it repaired? What will the cost be roughly? Will the lens ever work the same and have a good image quality? Should I ship it as is, or attempt cleaning it with air? Where is the best place to cry a little?   Thanks, any good info on this would be appreciated.  I will never rest a lens on another lens, I nearly lost a 24-70mm L that shattered a UV filter also.   Good reason to have a UV always, never thought it could happen to me.  Update: Looking at it more it looks like maybe just the front ring came undone and the UV filter broke?  I can't remember what it looked like exactly before.  I threw away the glass shards (polished sand) in the Nevada desert.      UPDATE: Primary reason for thinking more than just the UV was broken,   UV filter split into two rings, depositing the locking collar into the   lens barrel. urlEvent is undefined When should I fudge dice rolls? When I started my first D&amp;D campaign as a DM, I accidentally rolled and confirmed a crit on a PC from a goblin. It was an instant kill as our campaign had no cleric. That time I decided to fudge the dice. Now, there's a similar problem, but this time the PCs are at 3rd level and they are dealing with a very important NPC. If they fail this one diplomacy check, the entire game world will fall apart, as the king would then send his entire army against the PC's homeland and will destroy it, plunging my entire game world into chaos, at which time drow that I wanted to save for a later adventure will have to come out, which will instigate a wizard's guild to release their secret ultra-deadly virus, and everything in my game world, including the PCs, will die. So should I fudge the dice roll? When is it good to fudge it? The limit of a complex integral If $f(z)$ is continuous in the neighborhood of the point $z=a$, then $$\lim_{r\to 0^+} \int_{\vert z-a\vert =r} \frac{f(z) dz}{z-a}=2\pi i\,f(a)\tag{1}$$  In my desperate intents of proving the above exercise, I tried to prove a 'particular' case of it, which is when $z=0$.   So if $z=0$ and as $f$ is continuous on a neighborhood of zero, given $\epsilon&gt;0$, there exists $\delta&gt;0$ such that $|f(re^{i \theta})-f(0)|\le \epsilon$ for $r\in[0,\delta)$ and $\theta\in\mathbb R$. Hence,  $\left\lvert\int_{0}^{2 \pi}(f(re^{i \theta})-f(0))d\theta\right| &lt;2\pi\epsilon \Rightarrow \limsup_{r\to0}\left\lvert\int_{0}^{2 \pi}(f(re^{i \theta})-f(0))d\theta\right| \le2\pi\epsilon.$  If $\epsilon\to0$, then $\limsup_{r\to0}\left\lvert\int_{0}^{2 \pi}(f(re^{i \theta})-f(0))d\theta\right|=0 \Rightarrow \lim_{r\to0}\int_{0}^{2 \pi}(f(re^{i \theta})-f(0))d\theta=0.$  Is this proof correct?  How can I make the proof for $(1)$ following the 'particular proof'? Can someone help me? Why are these black dots / spots appearing in my pictures? Clusering based on categorical variables? I am working on a project and currently experimenting cluster analysis. The dataset is mainly categorical variables and discrete numbers. Please pardon my poor programming skills as I am not very familiar with MathJax, but I will try to summarize the data set in words in the following three examples.   imagine column 1 is participants names of course, from column2 - 5, each column's value range from 1 (least important) - 5 (most important). so in this case, column 2-8 only have discrete data.  column 6 for example, since this is a multiple choice question. row 1 chose "nice" as an answer, however, row 2 chose "poor". In this case, we have one column that contains multiple categorical answers. for column 7-9, this is another type of multiple choice questions. However this time, each column represents only one answer. column 7 only allows string values "true", column 8 only allows string "somewhat". So in this case, we have multiple columns that represent multiple answers of the same question.    Any ideas how to solve this? Thankful to any input!   A2       A3       A4       A5       A6       A7         A8         A9 1        4        5        4        nice     true       somewhate  false 2        4        3        1        poor     true                  false 1        5        2        1        nice                somewhate     3        2        1        5        nice     true                  false