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I have studied some of Einstein's Theory of General Relativity, and I understand that it states that gravity isn't a force but rather the effects of objects curving space-time. If this is true, then why are we instructed in middle school that it is a force? How do I search a specific user's answers? Where can the last FSTrim be seen as successful or not? (log files) I recently installed an SSD and AFAIK Ubuntu makes a weekly FSTrim on SSD's that support it.  Is there a log file of this trims? Entering Unicode characters in LaTeX How do I enter Unicode characters in LaTeX?  What packages do I need to install and what escape sequence do I type to specify Unicode characters in an ASCII source file? "INSERT IGNORE" vs "INSERT ... ON DUPLICATE KEY UPDATE" While executing an INSERT statement with many rows, I want to skip duplicate entries that would otherwise cause failure. After some research, my options appear to be the use of either:  ON DUPLICATE KEY UPDATE which implies an unnecessary update at some cost, or INSERT IGNORE implies an invitation for other kinds of failure to slip in unannounced.  Am I right in these assumptions? What's the best way to simply skip the rows that might cause duplicates and just continue on to the other rows? Ideals in the ring of Gaussian integers What are the proper ideals, prime ideals, maximal ideals of $\mathbb{Z}[i]$, the ring of Gaussian integers.    Check whether $(1+i)$ is prime or maximal ideal.     Can someone help me please. I have no idea how to crack this problem. Thanks for your time. Can someone explain Cayley's Theorem step by step? This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. Can someone explain this proof step by step? Thank you so much! Here is the proof:  Let $G$ be a group. We show that $G$ is isomorphic to a subgroup of $S_G$. Define a one-to-one function $\phi: G \to S_G$ such that $\phi(xy)=\phi(x) \phi(y)$ for all $x,y \in G$. For $x \in G$, let $\lambda_x: G \to G$ be defined by $\lambda_x (g) = xg$ for all $g \in G$. The equation $\lambda_x (x^{-1} c) = x(x^{-1} c) = c$ for all $c \in G$ shows that $\lambda_x$ maps $G$ onto $G$. If $\lambda_x (a) = \lambda_x (b)$, then $xa=xb$ so $a=b$ by cancellation. Thus $\lambda_x$ is also one to one, and is a permutation of $G$. We now define $\phi: G \to S_G$ by defining $\phi(x) = \lambda_x$ for all $x \in G$. To show that $\phi$ is one to one, suppose that $\phi(x) = \phi(y)$. Then $\lambda_x = \lambda_y$ as functions mapping $G$ into $G$. In particular $\lambda_x (e) = \lambda_y (e)$, so $xe=ye$ and $x=y$. Thus $\phi$ is one to one. It only remains to show that $\phi(xy) = \phi(x) \phi(y)$, that is, that $\phi_{xy} = \lambda_x \lambda_y$. Now for any $g \in G$, we have $\lambda_{xy} (g) = (xy)g$. Permutation multiplication is function composition, so $(\lambda_x \lambda_y)(g) = \lambda_x (\lambda_y (g)) = \lambda_x (yg) = x(yg)$. Thus by associativity, $\lambda_{xy} = \lambda_x \lambda_y$. "I want you succeeding" vs. "I want your succeeding" What does the dot mean On my reelbox (running Ubuntu) I have found a file in /etc/init that contains the following  # frontpanel-pre - check for frontpanel CAPs and adjust time #  description     "check frontpanel caps"  #start on starting mountall #start on tty-device-added DEVNAME=/dev/ttyS0  task  script         (         /sbin/dev_frontpanel.sh         /sbin/reelfpctl -capability         ) &gt; /dev/.frontpanel.caps         initctl emit --no-wait frontpanel-linked end script   I wonder if the dot in /dev/.frontpanel has some special meaning in linux  I thought the output of the commands in the brackets will be written to a file called ".frontpanel.caps" in /dev/ but there is no such file. In /dev/ there is a frontpanel which is a link to /dev/ttyS0  Could it be, that e.g. echo something &gt; /dev/.frontpanel.caps actually sends data (something in this case) to /dev/frontpanel ?  What does .caps do then? Isn't the uncertainty principle just non-fundamental limitations in our current technology that could be removed in a more advanced civilization? MySQL Results as comma separated list I need to run a query like:  SELECT p.id, p.name,         (SELECT name            FROM sites s           WHERE s.id = p.site_id) AS site_list   FROM publications p   But I'd like the sub-select to return a comma separated list, instead of a column of data.  Is this even possible, and if so, how? While reading a book, I came across the word I'd've, as in:     I'd've argued against it.   While it was obvious what it meant, it left me puzzled. Is I'd've a proper word? biblatex: Handling secondary ("quoted in") citations This question and the answer became part of a package:      How can I qoute secondary quotes which I’ve taken form another book.  Example  Jon Doe writes a book an cites Fred Bloggs in it. Now I read the book of Doe an want to cite his cite of Bloggs. So my bibliography entry must look like     Bloggs, Fred: A nice old book, 1850, p. 35. Quoted in Doe, John: A bad new Book, 2011, p. 67.   Doe’s book should not be listed separately in the bibliography.  I know that there’s a bibstring quotedin containing “quoted in” (p. 159 of biblatex’s manual) but I don’t know if theres a (predefined) way to use it …  Update  It seems that there’s no predefined way, so I decided to use userc for crossreferences like this. This works for the bibliography but not for cite commands. At the moment I create the “in text cite” manually.  It would be great to have a macro like  \QIautocite[&lt;prenote a&gt;][&lt;postnote a&gt;]{&lt;author a&gt;}[&lt;prenote b&gt;][&lt;postnote b&gt;]{&lt;author b&gt;}   that expands to the above given example.  That’s my new code, which gives the right results for bibliography and “in text cite” but the latter not automatically …  \RequirePackage{filecontents} \begin{filecontents}{\jobname.bib} @BOOKLET{autora,   title = {A nice old book},   userc={autorb},   author = {Fred Bloggs},   year = {1850}, } @BOOKLET{autorb,   title = {A bad new book},   author = {Jon Doe},   year = {2011}, } \end{filecontents}  \documentclass[ngerman]{scrartcl}  \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc}  \usepackage{babel,csquotes,blindtext}  \usepackage[%     bibstyle=authortitle,     citestyle=verbose-ibid, ]{biblatex} \addbibresource{\jobname.bib}  \NewBibliographyString{bibcitedas} \DefineBibliographyStrings{german}{%   bibcitedas = {{zitiert als}{zit. als}}, }  % Put the 'quoted in text' behind the bibentry. \renewbibmacro*{finentry}{%   \ifbibliography{%     \iffieldundef{userc}{}{%       \newunit\bibstring{quotedin}\addspace\fullcite{\thefield{userc}}     }   }{}%   \finentry% }  \begin{document} \blindtext \begin{quotation}     \blindtext\footnote{\cite[36]{autora}. Zit. nach \cite[89]{autorb}} \end{quotation} \blindtext\footnote{\cite[36]{autora}. Zit. nach \cite[89]{autorb}}  \printbibliography \end{document}   Example 2  In addition to my comment (Hi @Audrey. This works fine :-) but theres a little thing …) on Audrey’s answer:  If I use the quotecite command like this  \quotecite[1]{reese}[5]{weinberg} \quotecite[1]{reese}[5]{weinberg} \pagebreak \quotecite[1]{reese}[5]{weinberg} \quotecite[1]{reese}[5]{weinberg}   the footnotes should look like       1 Reese [full cite], p. 1. Zit. nach Weinberg [full cite], p. 65   2 Ebd.   ———— page break ————   3 Rese [short cite], p 1. Zit nach Weinberg [short cite], p. 65   4 Ebd.   Everything works like expected except footnote 3 which is     3 Rese [short cite], p 1. Zit nach Weinberg [full cite], p. 65     at the moment  Update 2  Since it seems not possible to react to autocite I created a new plain command with  \DeclareMultiCiteCommand{\quoteciteplain}[\cbx@qcwrapperplain]{\cbx@quotecite}{\cbx@qcdelim}  \newrobustcmd{\cbx@qcwrapperplain}[1]{%   \global\toggletrue{cbx@isquote}%   #1%   \global\togglefalse{cbx@isquote}%   \global\togglefalse{cbx@iflastibid}%   \global\togglefalse{cbx@ifquoteepost}%   \global\togglefalse{cbx@ifquoterpost}} It's using Euclidian inverse. Can you show it step by step?   $$N=7, q=11, a = 3+2X^2 -3X^4 +X^6$$  the inverse of $a \pmod {11}$ is  $$A=-2+4X+2X^2 +4X^3 -4X^4 +2X^5 -2X^6$$  How do we compute $A$? Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$ I'm having problems with an exercise from Apostol's Introduction to Analytic Number Theory.     Given $x$ and $y$, let $m=ax+by$, $n=cx+dy$, where $ad-bc= \pm 1$. Prove that $(m,n)=(x,y)$.   I've tried to give a proof, but I suspect it's wrong (or at least not very good). I would be very thankful for any hints/help/advice!  My proof:     We observe that since $ad-bc= \pm 1$, $ad=bc \pm 1$, and $(ad,bc)=1$.   Now, $(a,b) \mid m$ and $(c,d) \mid n$ but      $$(a,b) = ((d,1)a,(c,1)b)=(ad,a,bc,b)=(ad,bc,a,b)=((ad,bc),a,b)=(1,a,b)=1.$$      Similarly, we determine $(c,d)=1$. So, $1=(a,b) \mid m$ and $1=(c,d) &gt; \mid n$. But $(x,y)$ also divide $m$ and $n$. Since $(x,y) \geq (a,b)=(c,d)=1$, this implies that $(x,y)=m,n$. Hence   $(m,n)=((x,y),(x,y))=(x,y)$. How can I calculate the angle and proper turn direction between two 2D vectors? I am working on some movement AI where there are no obstacles and movement is restricted to the XY plane. I am calculating two vectors, v, the facing direction of ship 1, and w, the vector pointing from the position of ship 1 to ship 2.  I am then calculating the angle between these two vectors using the formula  arccos((v · w) / (|v| · |w|))   The problem I'm having is that arccos only returns values between 0° and 180°.  This makes it impossible to determine whether I should turn left or right to face the other ship.  Is there a better way to do this? Equals(=) vs. LIKE When using SQL, are there any benefits of using = in a WHERE clause instead of LIKE?  Without any special operators, LIKE and = are the same, right? How exactly do DLL files work? There seems to be an awful lot of them, but I don't know what they are or how they work.  So, what's the deal with them? How do American English and British English use the definite article differently? I decided to make sure that I know this important difference between American and British English, so I wrote what I have found out so far and I would be grateful to anyone who reads this and tells me whether I am wrong, or not.  In British English when people say  to hospital or in hospital when  talking  about somebody being there as a patient they don't use the definite article : "I had to go to hospital", "She spent two weeks in hospital". And the meaning is that somebody was there as a patient.   If then for some other reasons British English speakers  will use the definite article which will change the meaning itself, I noticed that, in American English, native speakers often use the  the and if they need to show that somebody is in church to pray,  in school as a student, in hospital as a patient, in prison as a prisoner, they use 'in' and not 'at'. Do American English speakers use 'at' like British English speakers use 'the' to give the sentences a different meaning?    Are my sentences correct? Do they show American English usage?   He is in the school. (enrolled as a student) He is at the school. (for some different reasons) He is in the hospital. (as a patient) He is at the hospital. (visiting somebody) He is in the church. (to pray) He is at the church. (for some different reasons) He is in the university. (as a student) He is at the university. (not as a student) He is in the college. (as a student) He is at the college. (Not as a student) He is in the prison. (as a prisoner) He is at the prison. (not as a prisoner) Which one of these two statements is correct?     Our staff do ...      Our staff does ...   And is staffs ever correct?