Build a bijection $f: \mathbb{R} \to \mathbb{R}\setminus \mathbb{N}$.  What about $f(x)=\pi \cdot x?$
Find a bijection from $\mathbb R \to  \mathbb R-\mathbb N$. I want to set my function up such that all natural numbers get mapped to $n+.1$ and reals of the form $n+.1$ to $2n+.1$ is this correct?
Java rounding to nearest 0.05 The temperature of an electron Does an electron have a temperature, if so, what is it?  Imagine an electron (Ke = 1 eV) in a tube at room temperature (300 K)    what is its temperature?   Imagine now same electron in space (3 K) with same Ke, is it any different from the other?  What is the influence of external temperature on electrons?  Do those electrons have electric energy, or do we have such energy only when the electron hits something and discharges its Ke? Why is the last digit of $n^5$ equal to the last digit of $n$? I was wondering why the last digit of $n^5$ is that of $n$? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ideas about how I can figure out the proof myself? Thanks. I'm using Windows 7 as the host machine with VMWare Workstation 9. I created a virtual machine Ubuntu Desktop 12.04 LTS 64 bit version. There are two options for log in, Ubuntu and Ubuntu 2D. Whenever I log in to Ubuntu, the task bar will not show up, is like having a black margin all around the desktop background and only the center is visible.  I wish I could attached a pic, but I need at least 10 reputation.  When I move my mouse over the left side of the desktop I can see that something is there but it is shown only as a black square. I tried deleting the virtual machine I created for Ubuntu and re-installed it, but it keeps appearing like that.  When I switched to log in with the Ubuntu 2D version everything worked alright. I could see the task bar, the icons, the clock etc. What might be causing this problem? "Which" vs. "what" — what's the difference and when should you use one or the other? Most of the time one or the other feels better, but every so often, "which" vs. "what" trips me up.   So, what's the exact difference and when should you use one or the other? Not getting email alerts on answers on Stack Overflow Not getting email alerts on answers on my questions on stackoverflow.com   Used to get it earlier (>1 month back), but not getting alerts in recent times Email field in profile is filled  In prefs, "I'd like email notifications of activity on questions I've asked" is ticked.   What is missing?  Is it possible that signing up on another Stack Exchange community (say, Server Fault), on which by default, "I'd like email notifications of activity on questions I've asked" is unchecked affects same preference on stackoverflow.com? If so, it is a bug. Prove that $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$ How many possible electromagnetic wavelengths are possible? Disclaimer: Please keep in mind that I am a young highschool student with no background in physics, this research was done in the course of an hour, and my reasoning could very well be wrong.  After a short conversation about color, I did some thinking, and read  The answer claims that there are $\infty^{\infty}$ colors   My first problem is the definition of colors, I'm going to simplify this (although it's not the same thing) as electromagnetic frequencies. This is what I mean when I refer to color in the rest of this question  The problem with $\infty^{\infty}$ colors is that the plank length exists. This causes two problems to arise with $\infty^{\infty}$. If $l_p$ is assumed to exist (as it is), this means the smallest distance in the universe is $l_p$. So, all wavelengths of light can be divided into a set, of the wavelengths produced by objects at $0^{\circ}$ Kelvin to $\infty^{\circ}$ Kelvin. According to this logic, there are simply $\infty$ colors.  Things are complicated further, when you introduce the Plank Temperature ($P_t$). $P_t$ is the theoretical limit temperature of the universe (at least for the purposes of light and color) Because an object hotter than $P_t$ will produce light with wavelengths shorter than $l_p$. This would introduce an upper limit on colors too.  Wouldn't this mean that the total number of possible colors is equal to however many plank lengths difference there is between an electromagnetic wave produced by a 0 degree object and a $l_p$ object? And therefor far less than $\infty$? Does this reasoning make sense at all? After trying to delete apache and tomcat, and now trying to reinstall apache 2, I am missing the init.d/apache2 file to start/stop my server.  I am not sure if the new install actualy worked, and my apt-get purge apache2 didn't remove everything in the first place!  How can I get the init.d/apache2 file to test if server starts / is installed properly (I had removed files and directories manually)? Is there a way to preserve a weapon in Magicka within a saved multiplayer game Give moderators the ability to mark reviews as "bad" From what I've seen, many SO moderators trawl completed reviews and watch out for robo reviewers; and manual banning them when they see enough (>= 3?) bad reviews. This is inefficient because:   It doesn't integrate into the audit system. There's no "memory" to it. If mod A finds that user X has made two (any number less than the threshhold) bad reviews, he may make a mental note but there probably won't be any permanent/shared memory of this. If moderator B finds this user at a later date, he will have to look through the entire history, including stuff that mod A has already checked.    What I propose is that moderators get the ability to mark individual reviews as "invalid" from the completed review page. This is only to be used for reviews where any reviewer who is paying attention (and knows the rules) would choose the opposite reason (spam and other low quality stuff).   These bad review flags are treated like audits for a user -- if the number of failed audits and bad reviews is enough, the user gets autobanned. The nice thing is that this system should (probably) easily be able to , fixing the "Why was I manual banned?" problem.  This could also be integrated well with . This is a homework question. $d,n\ge 2$.  Let $L=\{(x_1,...,x_n)\in (\mathbb{R}^d)^n: x_i\in \mathbb{R}^d,  x_i\ne x_j \forall i\ne j\}$.   I tend to think it is not path connected because if you assume $d=1$, this is the case. But I cannot prove for $d\ge 2$.  Any help will be appreciated! :) I've found strange behavior when closing this  - I definitely see that there are 3 closing votes and 2 of them are at "Off-topic" section:   -but there are no explanations on section itself:   -while, normally, it's expected that it will appear. In custom comments section there are also no closing-votes displayed.   Perhaps it's just my browser (which is FF 26 x64 under Win) - so marking this as a "possible bug". Thank you for your attention. "This edit would have to be a suggested edit" / edit button grayed out A couple of times recently when I attempted to edit a post while reviewing it from the "First Posts" queue, I received the following message when I tried to submit the form:  This edit would have to be a suggested edit. (x)   This is a rather cryptic message, considering that edits made from a review are suggested edits, subject to the same peer review process. There was no indication of how the edit would have to be submitted differently, much less why. I searched both help and meta for more information about this message, but came up empty.  The first time it happened, after I cancelled out I saw that someone else had just edited the post, so I thought maybe that's why I couldn't submit an edit from the review (and wondered why, if that's the case, the error message didn't just say that instead of telling me something that doesn't make any sense - that the edit I'm trying to suggest can't be submitted because it would have to be a suggested edit). However, the second time I received this message (last night, for this question: ), there was no other edit. So, I clicked the No action needed button and tried to edit the post outside the review process, only to discover that the edit link was grayed out.   I see several explanations of possible causes for graying out the edit link here: . However, there was no tooltip when I hovered over the disabled link, and the edit queue was in the single digits.  A few minutes later, I noticed that the edit link was no longer grayed out, so I edited the post and my edit was accepted.   My best hypothesis based on my experience with the error so far and the explanations I've read here at meta is that someone submitted a suggested edit while I was reviewing the post, which (a) caused me to get the cryptic error message and (b) caused the edit link to be grayed out when I accessed the post the normal way. Then a few minutes later, the suggested edit was rejected, opening up the question for new edit suggestions.  Rather than speculating about it and trying to determine the answer scientifically, it would be helpful if someone could give me the answer (as Seinfeld fans might put it, "Why don't you just tell me why I can't edit the post?"). It would be even more helpful if the error message itself provided the explanation, or if the message were defined somewhere in the help pages. But, at the very least, hopefully there will soon be an answer on meta to enlighten people who scratch their chins over this in the future. :) Is there a way to convert number words to Integers? How can I detect if my apple device supports Bluetooth Low Energy Argument list too long for ls I get the following error when trying to ls *.txt | wc -l a directory that contains many files:  -bash: /bin/ls: Argument list too long   Does the threshold of this "Argument list" dependent on distro or computer's spec?  Usually, I'd pipe the result of such big result to some other commands (wc -l for example), so I'm not concerned with limits of the terminal. Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$ Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$.  Ok I'm finally understanding what each of these things mean.. but I'm running out of ideas on how to actually show it without using arbitrary vector space examples like Rn. Since sets and subspaces are kind of new its hard to figure out how to write correct proofs with them.   I can see how if $V = \mathbb{R}^3$ and $S = \{(1,1,1)\}$ and $W = \{(1,1,1), (1,1,2)\}$ then $\mathrm{span}(S) = R1(line), span(W) = R2(plane)$ and that R1 is a subspace of R2... im just having trouble showing this officially when all vector spaces and sets are completely in general....  I could get as far as writing out $S=(u_1,u_2,\dots,u_n)$, $u_i \in V$ and same with $W$, but I can't really figure out what else to do in this sort of proof. I have a dual boot of Windows 10 and Ubuntu 16.04. Recently I upgraded it to 18.04. As soon as I did it, it started showing DISK IS LIKELY TO FAIL SOON (37° C / 99° F).    I checked the status of my disk via Windows and it shows it’s perfectly healthy. I assumed the error occurred because my partition size of Ubuntu was 64GB (which I now increased to 86GB). Please help me out with this.    Edit: The results of smartctl - attributes Taylor Expansion $\log(\frac{1+z}{1-z})$