C++ circular header includes
Resolve build errors due to circular dependency amongst classes
Can you help me resolve this paradox for a geodesic orbit? Does it maximize proper time to sit still rather than orbit for 1 year? If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$. Let $G$ be a group with $|G| = n$ and let $ \emptyset \ne S \subseteq G$.   I want to show that $S^n$ is a subgroup of $G$ where by $S^n$ I mean the set $\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$. Evaluation Symbol: Consistent Implementation Should my first trip be to the country which issued my Schengen Visa? In case, if I get a multiple-entry Schengen visa in one of the embassies in Ukraine, Kyiv - should my first trip be to the country which issued the visa?  Any there any requirements of this sort? Does this condition vary depending on embassy which issued the visa?  What are the possible consequences on not visiting that country first? Does it somehow depend on country where the embassy is (i.e. non-Schengen and not EU members)? What does a "CALLBACK" declaration in C do? I was looking through some code from the  library and came across a function declared like this:  LRESULT CALLBACK WndProc(HWND hwnd, UINT msg, WPARAM wParam, LPARAM lParam)   Now, I'm a Delphi coder.  No hablo C muy bien, senor.  But I remember enough syntax from my college courses to read it like this:  Function name is WndProc.  Argument list is pretty self-explanatory.  Function return type is LRESULT.  But what in the world is that "CALLBACK" doing there?  In Delphi, any function can be used as a callback; you just need to pass the right type of function pointer.  Is there any particular reason why C doesn't work that way?  Or does it mean something different? Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$ I am stuck with the following problem.     Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$   I was thinking of using the triangle inequality saying $$|a_n + b_n| \le |a_n| + |b_n|$$ but the problem is not about absolute values of the sequence.  Intuitively it's clear that this is true because $a_n$ and $b_n$ can "reduce each others magnitude" if they have opposite signs, but I cannot express that algebraically...  Can someone help me out ? Let $1, \omega, \dots, \omega^{n-1}$ be the roots of the equation $z^n-1=0$, so that the roots form a regular $n$-gon in the complex plane. I would like to calculate $$ \prod_{j \ne k} (\omega^j - \omega^k)$$  where the product runs over all $j \ne k$ with $0 \le j,k &lt; n$.    My attempt so far  Noting that if $k-j = d$ then $\omega^j - \omega^k = \omega^j(1-\omega^d)$, I can re-write the product as $$ \prod_{d=1}^{\lfloor n/2 \rfloor} \omega^{n(n-1)/2}(1-\omega^d)^n$$  I thought this would be useful but it hasn't led me anywhere.  Alternatively I could exploit the symmetry $\overline{1-\omega^d} = 1-\omega^{n-d}$ somehow, so that the terms in the product are of the form $|1-\omega^d|^2$. I tried this and ended up with a product which looked like $$\prod_{j=0}^{n-1} |1 - \omega^j|^n $$  (with awkward multiplicative powers of $-1$ left out). This appears to be useful, but calculating it explicitly is proving harder than I'd have thought.  The answer I'm expecting to find is something like $n^n$.    My motivation for this comes from Galois theory. I'm trying to calculate the discriminant of the polynomial $X^n+pX+q$. I know that it must be of the form $ap^n+bq^{n-1}$ for some $a,b \in \mathbb{Z}$, and putting $p=0,q=-1$, the polynomial becomes $X^n-1$. This has roots $1, \omega, \dots, \omega^{n-1}$, so that $(-1)^{n-1}b$ is (a multiple of) the product you see above. An expression for $a$ can be found similarly by setting $p=-1,q=0$. Fibonacci proof question: $f_{n+1}f_{n-1}-f_n^2=(-1)^n$ Does it make sense to Favourite a Question you created? Is this just a redundant act, or are there differences? Reload the page on hitting back button Bibliography with BibLaTeX: Do not abbreviate first given (first) name and abbreviate all further given names The first given name of an author or editor in my bibliography should be written out, but all further first names should be abbreviated (initials with a period -- and space if there are multiple first names/initials, see below). I am using style=authoryear with BibLaTeX (v. 2.7) and Biber (v. 1.7).  My bibliography looks like this:     Doe, John Arthur Kyle (2013). The secret life of John Doe.   It should look like this:     Doe, John A. K. (2013). The secret life of John Doe.   Minimal example:  example.tex  \documentclass{scrartcl}  \usepackage[ngerman]{babel} \usepackage[babel,german=quotes]{csquotes} \usepackage[backend=biber,style=authoryear]{biblatex} \bibliography{bibliography}  \begin{document}  \cite[123]{Doe} \printbibliography  \end{document}   bibliography.bib  @book{Doe,     title = {The secret life of John Doe},     author = {Doe, John Arthur Kyle},     year = {2013} }   (Worked with said versions of BibLaTeX and Biber, Texmaker (v. 4.0.3) and MiKTeX (v. 2.9.4813).)  I could not find a solution in biblatex.def and firstinits=true abbreviates all first names, including the first one. Samba authentication fails Booking a flight without passport info I have been looking to book a bundle (hotel and flight) with Expedia to Puerto Vallarta MX. When I get to the booking section it does not ask for my passport number. Do they ask for my passport after I book? Or later when the date gets near? My passport is currently being renewed and I want to book my hotel and flight before they sell out but I am afraid it'll ask me for my passport number. Can anyone inform me about this situation? Why would Louise telling the Chinese commander about his wife's last words change his mind? In the future in Arrival, the commander says that he changed his mind about going to war with the aliens because Louise called his private number and told him his wife's dying words. My question is what did Louise say to him (like in English) and why did it change his mind? Compute Taylor error of $10^{-5}$ for $\cos(\pi/20)$ How do I get started creating an Ubuntu Desktop App? I want to create an Ubuntu desktop app, specifically to replace Gwibber.  I can code in Java, Python and a few other technologies. Adobe Flex would have been an option until Adobe decided to suck some more and stopped support for AIR on Linux.  Note - I don't care about supporting other OS's. It will be a personal Ubuntu specific app. It can I open-source, I don't mind extra help.  It should integrate seamlessly with Ubuntu, notifications and all.  I guess Python it is, is that correct? How do I get started? I was surprised to discover my dictionary had this entry for dilemma:     a situation in which a difficult choice has to be made between two or more alternatives, esp. equally undesirable ones   The notion of dilemma meaning two or more flies against what I was taught about the word. The very idea of a  is specifically based on the number two.  Has my dictionary merely updated its definition to encapsulate the many people who use dilemma for more than two equal choices? Or was someone in my youth being unnecessarily pedantic? Self-promotion in chat We know from the  that it is accepted only in a limited way in answers and frowned upon if found to be spammy. We know from  that it can be deleted at any time from comments if the comment is not relevant or spammy.  But what about the chat? This leaves chat to be the only legitimate place to promote products or services that the user is affiliated to (albeit in an ad-like way).  What is your opinion? Enumerate within TikZ node avidemux problems with Ubuntu 16.04 LTS I installed 16.04 a few days ago and had to reinstall some apps. One of them was avidemux. After I installed it from the Software Center, I couldn't get it to work properly. It wouldn't even load a video clip before giving me an error message and closing.   After trying many commands to purge the avidemux files, I still can't get it to reinstall from the Software Center...   Any tips?