Why isn't the background colour showing up in my divs?
What methods of ‘clearfix’ can I use?
Minimize multi-variable function one variable at a time I am wondering if I can minimize a multi-variable function one variable at a time. In other words, is it true that:  $min_{x_1,x_2} f(x_1,x_2)=min_{x_1} min_{x_2} f(x_1,x_2)$ What is a glue record? This is a  about DNS glue records.   What exactly (but briefly) is a DNS glue record?  Why are they needed and how do they work? Basically, I would like to make it so that the lighter a pixel is (the more white it has) the more transparent it becomes. Black pixels would be opaque, white pixels would be transparent.  I currently use the slider in blending options on the first tab to get rid of the white and make it transparent but find it works slightly better if I use the magic wand tool with delete.  I figure that there is some blending option out there but there are so many and I'm not sure which to use to achieve my goal.  (Using Photoshop CS6) How can I use an align environment flush left? Is there a way possible to imitate the align environment for math mode commands, without having all writing being placed toward the center on the output page?  MWE:  \documentclass{article} \usepackage{mathtools}  \begin{document} \begin{align*} \cos\theta_1 \cos\theta_2-\sin\theta_1\sin\theta_2 &amp;= \cos(\theta_1 +\theta_2) \\ \sin\theta_1 \cos\theta_2 + \cos\theta_1 \sin\theta_2 &amp;= \sin(\theta_1+\theta_2) \end{align*}  \end{document} Excuse me if the question is naïve. I am not a physicist.  Let's imagine formation of a black hole from a homogeneous collapsing star. At certain moment when enough of matter is inside of a small enough spherical area the event horizon starts to exist. When does the singularity start to exist and why?  The matter below the event horizon still continues to collapse towards the centre (e.g. in the reference frame of the falling matter). What are the necessary conditions for the singularity to appear?  I have found two similar questions but the answers did not really explain this problem to me: Having some trouble proving this:     If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show: $\lim _{x\to \infty } (f(x+1)-f(x))=0$   Attempt: from $\int \lim _{x\rightarrow \infty } f^\prime(x) \mathrm{d}x=0$ we can say that $f^\prime(x)=const$ would we then have to show that  $\lim _{x\to \infty } (f(x+1))$ is also a constant? From there if we show that both are equal then the difference is zero. I am on the right track?  Thank you for the help. What are Virtual Methods? Why would you declare a method as "virtual".   What is the benefit in using virtual? I have a point $(X,Y)$ on an elliptical curve $E(a,b)$ where $a=-3$ and $B$ is a large number that is in hexadecimal from -51BD. To compress this point oficially in a program, we know that every $X$ on the curve has two $Y$'s, one even and one odd. Therefore, we only need to store whether the corresponding $Y$ point we are storing is even or odd. So to store the point $(X,Y)$ now, we shift $X$ ridding of its most significant bit and add $1$ if the $Y$ point is odd. Therefore, one can think of the compressed point being $2 \cdot X+B$ ($B$ is the bit we added to tell if $Y$ was odd or even) since the multiplication by $2$ is implied because of the shift. I understand this completely. It's recovering the original point that has me confused.  Since a point on the curve is given by the equation $y^2=x^3+ax+b$, we can find square roots officially in $\mathrm{GF}(p)$. The following from my spec is what confuses me.  If $p$ is prime and congruent to $3$ mod $4$, one of its square roots is $z^{(p+1)/4} \bmod p$ and the other is $p-z^{(p+1)/4} \bmod p$. How would I solve this? What is $z$?  For example, I know how to extract the $x$ point so let's say I simplify the right side of the equation to be $7$ and $p$ is $11$. Would I be trying to solve for the square root that's equal to $7$ mod $11$? How should a physics student study mathematics? Note: I will expand this question with more specific points when I have my own internet connection and more time (we're moving in, so I'm at a friend's house).  This question is broad, involved, and to some degree subjective.  (I started out as a physics-only student, but eventually decided to add a mathematics major. I am greatly interested in mathematics; the typical curriculum required for physics students is not deep or thorough enough; mathematics is more general (that means work!); and it only requires a few more classes. Naturally, I enjoy  immensely.)  This question asks mainly of undergraduate-level study, but feel free to discuss graduate-level study if you like.  Please do not rush your answer or try to be comprehensive. I realize the StackOverflow model rewards quick answers, but I would rather wait for a thoughtful, thorough (on a point) answer than get a fast, cluttered one. (As you probably know,  produces clear, useful writing; and a properly-done comprehensive answer would take more than a reasonable amount of time and effort.) If you think an overview is necessary, that is fine.  For a question this large, I think the best thing to do is focus on a specific area in each answer.    Update: To Sklivvz, Cedric, Noldorin and everyone else: I had to run off before I could finish, but I wanted to say I knew I would regret this; I was cranky and not thinking clearly, mainly from not eating enough during the day. I am sorry for my sharp responses and for not waiting for my reaction to pass. I apologize.  Re: Curricula:   Please note that I am not asking about choosing your own curriculum in college or university. I did not explicitly say that, but several people believed that was my meaning. I will ask more specific questions later, but the main idea is how a physics student should study mathematics (on his or her own, but also by choosing courses if available) to be a competent mathematician with a view to studying physics.  I merely mentioned adding a mathematics major to illustrate my conclusion that physics student need a deeper mathematical grounding than they typically receive.   And now I have to run off again. Why does the Prime Directive not apply to Rubicun III? In a first season episode of Star Trek: The Next Generation the Enterprise visits a planet named Rubicun III where () the Edo live.  It is often mentioned in Star Trek that the Prime Directive requires that pre-Warp civilisations be ignored until they discover Warp mechanics. But this planet and civilisation appears to be exempt from that status. Why? as of 4/2015 OS X Kindle App: Where are the books located? After I updated my Kindle for Mac app 2 mos. ago I can no longer find the azw book files on my Mac to convert them so I can read them on my KOBO ereader. I use to find them in a My Kindle Content file but that has disappeared. I have Ult. DRM &amp; Calibre but they won't work without the azw files, and I can't find them on my mac anymore. Can anyone help me? The answers on this site about this are years old &amp; do not apply to my problem. Georeferencing scanned map using ArcMap? Is wildcard search possible in Gmail? I want to limit some users daily computer usage on the family PC. Currently we only have one user account set up on it, but I could create new ones. So Is there a way to set daily time limits for user on a windows 7 PC? Ideally I would like to set up it up for 2 hours on weekdays and 3 hours on weekends. Creation and annihilation operators in the hydrogen atom Why do most 2D platformers start off in the left and progress in the right direction? In almost all 2D platformers I've played, your avatar always starts off on the left side of the world, and continues on to the right. Is there something designers gain by doing this? The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$ It is well-known that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the prime ideals of $S_{(f)}$, the subring of $S_f$ comprising the homogeneous elements of degree $0$.    This is proposition II.2.5b in Hartshorne, exercise 5.5B in Ravi Vakil's notes $[1]$ (p. $130$ of the February $24$, $2012$ version), and proposition 8.1.21 of Akhil Mathew's notes $[2]$ (p. $136$). Unfortunately I cannot follow any of those proofs to my own satisfaction, perhaps because I'm not well-versed in commutative algebra.  The crux of the proof appears to be to show that, given a homogeneous prime ideal $\mathfrak{p}$ of $S$ not containing $f$, the construction used to obtain a prime ideal $\Psi (\mathfrak{q})$ of $S$ from a prime ideal $\mathfrak{q}$ of $S_{(f)}$ will recover $\mathfrak{p}$ when $\mathfrak{q} = S_f  \mathfrak{p} \cap S_{(f)}$. To be precise, let $\Psi (\mathfrak{q})$ be the homogeneous ideal of $S$ generated by $$\bigcup_{d \in \mathbb{N}} \{ s \in S_d : s / f^d \in \mathfrak{q} \}$$ and let $\Phi (\mathfrak{p}) = S_f \mathfrak{p} \cap S_{(f)}$. It's easy to see that $\Phi \circ \Psi$ acts as the identity on $\operatorname{Spec} A$ (or, for that matter, the set of all ideals of $A$), but I cannot see any obvious reason why $\Psi \circ \Phi$ should act as the identity on the set of prime ideals of $S$ not containing $f$. A detailed proof of this point would be much appreciated.  References  $[1]$ .  $[2]$ Algebraic geometry notes (covers material at the level of the first and second volume of EGA): , and . I'm new to Linux but not to app development. I come from a RealBasic on a Mac background.   I've accepted that I need to learn a new language to develop for Linux (since RB won't run on 64 bit distros and is a cloed proprietary language) but I want to make sure I pick the right one.   I've been looking at Python and was all set to go until I read that PyGTK is becoming obsolete with the introduction of GTK 3. Is this true? The Ubuntu developer website stills seems to recommend it with Glade for the UI but I want to make sure that I don't learn something that's on the way out! I'm just getting started with Pinter's A Book of Abstract Algebra, please be kind. The book solution for Chapter 2, Problem 7 claims that the following operator is non-associative:  $$ x * y = \frac{xy}{x+y+1}$$  The book solution:  $$ (x * y) * z = \left(\frac{xy}{x+y+1}\right) * z = \frac{xyz(x+y+1)}{x+y+z+xy+yz+xz+1}$$  $$ x * (y * z) = x * \left(\frac{yz}{y+z+1}\right) = \frac{xyz(y+z+1)}{x+y+z+xy+yz+xz+1}$$  However the $(x+y+1)$ and $(y+z+1)$ terms appear to actually cancel out:  $$ (x * y) * z = \left(\frac{xy}{x+y+1}\right) * z = \frac{\left(\frac{xy}{x+y+1}\right)z}{\frac{xy}{x+y+1} + z + 1} $$  $$ x * (y * z) = x * \left(\frac{yz}{y+z+1}\right)  =\frac{x\left(\frac{yz}{y+z+1}\right)}{x+\frac{yz}{y+z+1} + 1} $$  So the denominators cancel out, the two groupings are equivalent, the operator is associative.  This result is duplicated in Mathematica, but I'd like to make sure I'm not missing something.  Sorry for bothering people with such a basic question. "Can/may/will you help me with this?" Which word to use when we ask for help?  Some conditions:     We know that the person asked is able to do it. We don't know if the person asked is able to do it.