There is a solution which starts by taking $\langle 2+i\rangle$ as a subset of $\langle a+ib\rangle $ and then proceeds as, Hence $(2+i)$ can be represented as multiple of $(a+ib)$. I understand till this part.  Next it proceeds, $(2+i)=(a+ib)(c+id)$. I don't understand this line.  Shouldn't it proceed by multiplying $(a+ib)$ with an integer.  Why is $(a+ib)$  multiplied with another complex number?
I have to find an easy ring which this ring is isomorphic with.  I can prove that $5 = (2+i)(2-i) \in (2+i)$ but this doesn't help me further.  My book says the answer is $\mathbb {Z_5} $
biblatex: separator for citing multiple sources, some of them with page, some not I'm using biblatex for my references.  Normally, if I use \cite{source1, source2, source3}, I get a citation like  [3, 45, 98]   If I now want to give a page number for one of them, it looks confusing imho, as I should get (did not try it out up to now, honestly)   [3, 45, 78, p. 23f, 98]   So imho it would make sense to change one of the separator from a comma to e. g. a semicolon like:  [3; 45; 78, p. 23f; 98]   Question 1: Does that make sense, or are there other conventions for citing sources with and without page numbers together?  (Up to now I collected the citations without page number in one cite command and added singular \cite[][p. xy]{} commands for those with page numbers., so I'd get:   [3, 45, 98][78, p. 23f]   )  the \cites[]{}[]{}[]{} could also do that, but it also uses the same separators for entries and postnotes.  Question 2: How could I change the separator globally for those number-citations from comma to semicolon in biblatex? I am programming a tile based game and I have some basic tiles (grass, dirt, etc..), but I can't figure out how to make good random map generation, because when I do some really random selection, if the tile should be grass/dirt, I get this:    I understand why this is happening, but what I want is to create some random continuous areas of grass or dirt. Something that would make more sense, like this: Suppose $f$ is a differentiable function over $\mathbb{R}$ satisfying $f(0) = 0$. Show that if $f'$ is strictly increasing, then $\frac{f(x)}{x}$ is increasing over $(0,\infty)$.   Attempt:  Since $f'$ is increasing we know that $f'' &gt; 0$ for all $x$. Now let $g(x) = \frac{f(x)}{x}$ and thus $f(x) = xg(x)$. Then we wish to show that $g'(x) &gt; 0$ on $(0,\infty)$ and so since $f'(x) = xg'(x)+g(x)$ and $f''(x) = xg''(x)+2g'(x) &gt; 0$, we have $g''(x) &gt; \frac{-2g'(x)}{x}$. I am not sure how to use this though to prove the statement. How do I unload (reload) a Python module? I have a long-running Python server and would like to be able to upgrade a service without restarting the server. What's the best way do do this?  if foo.py has changed:     unimport foo  &lt;-- How do I do this?     import foo     myfoo = foo.Foo() We are given two independent standard normal random variables $X$ and $Y$. We need to find out the M.G.F of $XY$.  I tried as follows : \begin{align} M_{XY}(t)&amp;=E\left(e^{(XY)t}\right)\\&amp;=\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(XY)t}f_X(x)f_y(y)dxdy \\ &amp;=\dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(xy)t}e^{\frac{-x^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{-\dfrac{(x-ty)^2}{2}}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{-y^{2}(\frac{1}{2}+t^2)}dy \\ &amp;= \dfrac{1}{\sqrt{1+2t^2}} \end{align} Is this correct ? $$ A $$  Light is travelling away from the absolute center of the Schwarzchild black hole (i.e. none of that spinning/stretching field) at $\ C $, and so shouldn't $\ \Delta C$ need to be greater than $\ C $ in order for the light to collapse into the black hole, therefore violating the relativistic limit of $\ C $?    It seems like light would just 'hover' right in the center of a black hole then. Because if it goes away from it, the event horizon pulls it back, and if it goes into it, the velocity change must be greater than $\ C $ which violates general relativity; so technically there's no way for the light to move away, and so it would just be still.  The thing is that the light must move at the speed of light or general relativity doesn't work. It's massless... But in the center of a black hole, if the above is true (it can't move or escape), then it is not travelling at the speed of light. So it cannot be still.  It cannot be spinning in a constant orbit either, because that would require velocity in an outward angle. And it cannot move in an outwardly fashion, only in an inwardly one.  Is it spinning in an infinitely decaying spiral towards the center?  If so what happens to it if and when it reaches the direct and complete center, i.e. The case of it being 'shot' from the center.    $$ B $$  If you shot light at an angle towards a black hole, but not directly to the center of it  then that light from the outside will eventually become the same as the light that I said was shot from the center to the outside; because in both cases, they end up in an infinitely decaying spiral.    If so it would end up exactly like the light in the above paragraph, so then we need to figure out what happens to that light to figure this one out, or vice versa.    $$ C $$  If you shot light directly towards the center of a black hole it would have no spiral to decay into, so therefore it must stop at some point in the center of a black hole. If it stops though, it loses it's momentum and violates general relativity again because it's energy "disappears".    If so then it violates conservation laws. So I'm assuming this simply is not true.    Is the light travelling into a black hole somehow still travelling at the speed of light, in every direction possible where that direction keeps it facing towards the center?  If so, why is there not a point at which the only thing left for the light to do is stay still, since all other paths would increase the distance between it and the black hole's center, and therefore become Case A. And if there isn't, what would happen if there was?  Also, if it's a time and observation answer where it supposedly takes and infinite amount of time for C to hit the center, wouldn't it be the case that the black hole is 'eating' all of the available light in the universe and keeping it in stasis?  Also, what happens in a Kerr black hole then? Does it just rip the field of time/gravity at an angle such that the light would never get closer than the center? Proof verification of a number theory problem involving sequences. Agreement With Compound Subjects Joined by And understanding Debian dpkg package version numbers Wave equation: $y=A \sin(\omega t-kx)$ or $y=A\sin(kx-\omega t)$? What is correct wave equation: $y=A \sin(\omega t-kx)$ or $y=A\sin(kx-\omega t)$?  How are these wave equations used in the positive $x$-direction and negative $x$-direction? If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+...+a_n} = +\infty $ I've tried to use the limit test: $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = C$ in the series of $a_n$ and $\frac{a_n}{a_1+a_2+...+a_n}$ but it was inconclusive.  The question does not especify the sign of $a_n$ but proving it for positives would already help.  I think the sign can change as long as the partial sums never get to be zero. Like if the series has only negative numbers the result seems to stil hold. How to resolve ambiguous column names when retrieving results? I have two tables in my database:  NEWS table with columns:   id - the news id  user - the user id of the author)   USERS table with columns:   id - the user id   I want to execute this SQL:   SELECT * FROM news JOIN users ON news.user = user.id    When I get the results in PHP I would like to get associative array and get column names by $row['column-name']. How do I get the news ID and the user ID, having the same column name? Change name on the main Stack Exchange profile I use XeTeX, Version 3.1415926-2.4-0.9998 (TeX Live 2012/W32TeX).  I am using the crop package to add bleed and crop marks to a document. This works fine for pages with white background, but how do I achieve this for pages which have another colour in the bakckground?  I tried the pagecolor package, and see that it somehow extends the pagecolour to around 2 to 3 mm into the bleed but not beyond. How do I make the entire bleed region of the same colour as the page?  \documentclass[12pt]{article} \usepackage[dvips=false,pdftex=false,vtex=false,paperwidth=138mm,paperheight=216mm]{geometry} \usepackage[noaxes,noinfo,cam,dvips,pdftex,center,width=168mm,height=246mm]{crop} \usepackage{xcolor} \usepackage[english]{babel} \usepackage{blindtext} \begin{document} \pagecolor{orange} \blinddocument \end{document} How to type "ç" when using USA intl as the keyboard layout? I asked  but I can't make that solutions work on Unity. I want to use the keys ´ followed by c and the output should be "ç". It works that way in Windows out of the box, and it did work in Ubuntu 10.10 following the solutions given in the question above. But in 11.04 it outputs "ć". How to change that? How to draw arrows between colourboxes How do you "OR" criteria together when using a criteria query with hibernate? Mac asking for password for some files when deleting This happens not for all files. When I move or delete some of the files on my external drive Finder will ask for password. Is it due to permission set on these files ? Can I use something like Sudo when running Finder to prevent these pop ups ? The reason I'm asking is that sometimes I leave the mac running to move some files from one disk to another, just to find it on the password prompt a few hours later. Thank you. How do I create silhouettes in a studio environment? The trick to creating silhouettes in the field, as far as I know, is to use spot-metering and to expose for the background. However, this is a bit more difficult in a studio environment. What is the key to creating "artificial" silhouettes?  One solution that I'm considering is placing a bright flash directly behind the subject - would that work? How to train students to write high-quality research papers? As someone supervising postgrad students, I wonder what is the best way to train these students to write high-quality research papers?    The best way seems to be through coauthorship but ...  On the one hand I can be hands-off and rather give them feedback as best as I can but let them do most of the work themselves, sort of like a "sink or swim" strategy.  On the other hand I can be hands-on and rewrite the paper (or large parts of it) and hope that they learn from the example I give.  On the one hand I want them to learn to write papers without me some day and I want them to learn as quickly as possible to take load off me.  On the other hand I want us to publish in the best place possible with as little stress as possible.  On the one hand I want them to see the peer review comments from what they themselves have written.  On the other hand I do not want them to be discouraged by the peer review system early on and do not want to submit papers I know we could have written better.  On the one hand I want them to get confidence by being published and presenting work at good venues.  On the other hand I wonder if they need to get there themselves to get that confidence boost.    At the moment I'm very hands on. There's a few factors for this: I don't like submitting papers that are not up to a certain standard, I'm a native English speaker in a non-English speaking country with non-native speaking students, publications are a high priority for me, etc.  But recently a friend/colleague warned me that I should not be so hands-on with the writing of the papers of my students. He pointed to recently graduated researchers in my area that published good work as students but who struggled after leaving their PhD supervisor saying their supervisor did them no favours in the long term ... that by being so hands on, while the students have success getting published and so forth with me, I'm not really doing the most important thing a good supervisor should do: making myself obsolete.    My question is how to train students to write good research papers and in what ways can I balance being hands-on and hands-off?  I'm also interested in methods not directly involving coauthorship.