For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$
For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$
macro definition containing #include directive Is there a way to define a macro that contains a #include directive in its body?  If I just put the "#include", it gives the error  C2162: "expected macro formal parameter"   since here I am not using # to concatenate strings. If I use "\# include",  then I receive the following two errors:  error C2017: illegal escape sequence error C2121: '#' : invalid character : possibly the result of a macro expansion   Any help? How do high traffic sites service more than 65535 TCP connections? Can we have a high-speed low-drag close vote review queue? Sum of $k$-th powers How can I make a logo in Adobe Illustrator like the AntiParticle logo below, from ?   It is a large number circles of different, varying sizes, which form the shape of a text character. I know java and would normally put in getter/setter methods. I am interested in doing it in C# with the following code, but it throws a StackOverflow exception. What am I doing wrong?  Calling Code  c.firstName = "a";   Property Code    public String firstName; {     get     {         return firstName;     }     set     {         firstName = value;     } } I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. Thank you very much What does 5' and 3' mean in DNA and RNA strands? What are 5' and 3' in DNA and RNA strands? Please clarify with some images and please use simple English. Ubuntu 14.04 slow wireless connection Assume that $f$ be continuous on $\mathbb{R}$, $f'(x)$ exists for all $x\neq 0$, and $\lim_{x\rightarrow 0} f'(x)=1$. We need to show $f'(0)$ exist and is equal to $1$.  $f'(0)=\lim_{x\rightarrow 0}\frac{f(x)-f(0)}{x}$, $\lim_{x\rightarrow 0}f'(x)=1\Rightarrow\lim_{x\rightarrow 0}\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=1$...am I going in the right direction? Please help. How to determine the cause of MacBook battery not charging? I see that the question &quot;why is my MacBook not charging?&quot; gets asked a lot. The problem is, there are multiple reasons. Possible points of failure are: charger, cable, magsafe connector, DC input board, logic board, battery. (Have I missed any?). What is the best sequence for a user to follow, to diagnose the problem? By way of example, here are the first steps that I usually try:  Make sure the charger is working. Either use your charger on a different Mac, or test a spare charger with your Mac.  Check the connection. There might be gunk in your port that's stopping the connector from lying flat.  Try resetting the SMC and PRAM.  If the computer boots, run Coconut Battery to determine battery status.   This is where I get a bit stuck. For example, I have a machine here that boots fine when plugged in, but not from battery. Disconnecting the MagSafe causes the machine to switch off. Sounds like it's a dead battery, except that Coconut reports the battery as detected, in good condition, but not charging. What steps should I take next? Specifically, I'm asking for a generic algorithm to help determine the source of the fault (preferably without swapping out lots of components one by one)? Why can I no longer ask questions when all I did was ask some simple questions? I have asked the three questions below, one of which I deleted myself. How can Stack Overflow  block me from asking further questions? It seems Stack Overflow is not for beginners, but just for people with years of experience.   Could anyone tell me what the problem is with my questions and how I can improve them?      (deleted by myself) Why can't I import a SVG file from Adobe Illustrator or Inkscape? Converting expressions to polynomial form In Avengers: Infinity War, up until Thanos      gets all six Stones   in order to use the Gauntlet he needs to close his fist, as we see in the fight scene on Titan.     Doctor Strange: "Don't let him close his fist!"   So why later does snapping work? What word means what many people think 'ironic' means? 'Ironic' is often used to mean an unusual coincidence rather than its true meaning which is closer to sarcastic. That being said, is there a word that would be a good replacement for what many people seem to think it means? Printing the value of a variable in SQL Developer I'm editing a book that includes about 25 papers written in LaTeX by different authors.  Since many of them use the same \label, to avoid collisions I redefined the \label command (and \ref and \pageref and \cite and \bibitem)  so that if the third paper has  \label{main_theorem} the aux file actually uses the tag "iii:main_theorem", and \ref{main_theorem} actually refers to the same tag.  I did a similar thing for \cite and \bibitem.  %*************** bib/cite hacks \let\TheReal@citex=\@citex \let\TheReal@bibitem=\@bibitem \let\TheReal@lbibitem=\@lbibitem \gdef\@lbibitem[#1]#2{\TheReal@lbibitem[{#1}]{#2\roman{chapter}}} \gdef\@bibitem#1{\TheReal@bibitem{#1\roman{chapter}}} \gdef\@citex[#1]#2{% \toks@={}\count@=0% \@for\@citec:=#2\do{% \advance\count@ 1 \ifnum\count@=1\edef\0{\the\toks@\expandafter\expandafter\noexpand\@citec\roman{chapter}}% \else\edef\0{\the\toks@,\expandafter\expandafter\noexpand\@citec\roman{chapter}} \fi% \toks@=\expandafter{\0}}% \TheReal@citex[#1]{\the\toks@}% } %************** label/ref/pageref hacks  %%%%%%%%%%%%% must be before ams pkgs \let\TheRealLabel=\label \let\TheRealRef=\ref \let\TheRealPageref=\pageref \gdef\label#1{\TheRealLabel{\roman{chapter}:#1}} \gdef\ref#1{\TheRealRef{\roman{chapter}:#1}} \gdef\pageref#1{\TheRealPageref{\roman{chapter}:#1}}   This all worked fine until I used hyperref, which seems to ignore the redefinition of \label, whether I load it before or after the above code.  Any suggestions?  It would be nice if the redefinition would work whether or not hyperref was loaded, so I don't want to just grab the defintion of \label out of the package and tweak it.  I'm also open to alternative methods.  Redefining \label and \ref just seemed the most straightforward (to me) solution.  Note that I also want to be able to refer to the inner labels from outside, as well.  For example, I might want to say "the main theorem of the third paper is  \ChapRef{iii}{main_theorem}  from within the introduction (a different chapter). This is my last homework problem and I've been looking at it for a while. I cannot nail down what is wrong with this proof even though its obvious it is wrong based on its conclusion. Here it is:     Find the flaw in the following bogus proof by strong induction that   for all $n \in \Bbb N$, $7n = 0$.      Let $P(n)$ denote the statement that $7n = 0$.      Base case: Show $P(0)$ holds.      Since $7 \cdot 0 = 0$, $P(0)$ holds.      Inductive step: Assume $7·j = 0$ for all natural numbers $j$ where $0 \le j \le k$ (induction hypothesis). Show $P(k + 1)$: $7(k + 1) = 0$.      Write $k + 1 = i + j$, where $i$ and $j$ are natural numbers less than $k + 1$.  Then, using the induction hypothesis, we get $7(k + 1) = 7(i + j) = 7i + 7j = 0 + 0 = 0$. So $P(k + 1)$ holds.      Therefore by strong induction, $P(n)$ holds for all $n \in \Bbb N$.   So the base case is true and I would be surprised if that's where the issue is.  The inductive step is likely where the flaw is. I don't see anything wrong with the strong induction declaration and hypothesis though and the math adds up! I feel like its so obvious that I'm just jumping over it in my head. Conky Widgets are Opening in their Own Windows I just upgraded to 16.04 and I'm getting all my customization set up again, but it seems that most Conky widgets are opening in their own window, instead of just appearing on the Desktop; although some are working correctly.  For example:  LSD (working properly):    Seamod (not working, opening in own window):    Since it's not affecting every single widget, I assume it could be an issue with the conkyrc with the widgets that aren't working, but this also never happened in 14.04 so I wasn't sure if maybe there was some compatibility issue in 16.04. My Google-fu has failed me in finding a solution.  If anyone can point me in the right direction, it would be much appreciated! :)