Why aren't gas planets and stars fuzzy? The edge of Jupiter .    Even more bothersome, the  looks sharp, aside from kind of a soup of particles floating above it.    The sun's surface has an incredibly low density.  I mean, $10^{-6} g/cm^3$ kind of density ().  The Earth's atmosphere looks very hazy, and rightly so, because it's a gas.    Since Jupiter and the sun are 100% gas, why don't they appear as smudges on our telescopes?  To say "smudge" in a more scientific way, why isn't the transition zone between opacity and transparency a significant fraction of the radii of gaseous bodies?
Why is Jupiter so sharply defined? In photographs of Jupiter, . Being a gas giant, my naive self thinks that the atmosphere should have a more gradual cut off, creating a hazy effect similar to that on Titan. I am aware of Jupiter's immense size, so each pixel in the linked photograph is likely tens or hundreds of kilometers. Even so, wouldn't there be some small smudge? Does perhaps Jupiter's terrific gravity have any effect on the apparent edge of its atmosphere?
How can metagaming retries on poor skill checks be prevented? I'm sure every DM has had this happen: one PC is trying to find something cool on a dead monster's body. The player rolls poorly, and instead of accepting the paltry 2 copper pieces, the PC calls all the party members over to also try looting the corpse. Invariably, a different player rolls well, and everyone walks away satisfied that everything's been taken down to the poor dead monster's last holey sock.   Another example: One PC is trying to find some useful information in a library. There're no guards, no reason anyone would be bothered, and no time restraints to stress about. The player rolls poorly, so all the other PCs try, too.  I'd describe this as well-intentioned metagaming. The players are afraid of losing out on content, this despite multiple sessions of me making it clear that I don't structure my games to penalize gameplay that way.  These are situations that require a roll to determine the degree of success. (At least I think so; perhaps I'm wrong here?) Failure could set back the PCs a cool item or a bit of information that would have helped a little, but these minor failures aren't game-enders or anything. As a DM I can't see a reason to say "No" and disallow the practice, but it feels vaguely like cheating to let the players roll multiple times to get better results.  How do I prevent players ganging up on skill checks after they've seen someone roll poorly? Unicode Regex; Invalid XML characters The list of valid XML characters is well known, as defined by the spec it's:  #x9 | #xA | #xD | [#x20-#xD7FF] | [#xE000-#xFFFD] | [#x10000-#x10FFFF]   My question is whether or not it's possible to make a PCRE regular expression for this (or its inverse) without actually hard-coding the codepoints, by using Unicode general categories. An inverse might be something like [\p{Cc}\p{Cs}\p{Cn}], except that improperly covers linefeeds and tabs and misses some other invalid characters. Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$   Here is what I tried:  First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$.  Let $r$ be a root of $P$ (i.e. $P(r)=0$). It follows that all terms in the infinite sequence $r,r^2,r^4,\dots,r^{2n},\dots$ are roots of $P$. To avoid a polynomial with infinite roots, we have to have the sequence be periodic. So it turns out that either $r=0$, or the roots of $P$ are roots of unity of degree $2n$ for some $n\in\mathbb{N}$.  In the first case, we get $P(x)=xQ(x)$. Plugging in, we get $$x^2Q(x^2)=x^2Q(x)^2\implies Q(x^2)=Q(x)^2 ,$$ i.e. that $Q$ satisfies the same condition as $P$.  In the second case, we have $P(x)=(x^{2n}-1)Q(x)$. Plugging in, we get $(x^{2n}-1)Q(x^2)=(x^{2n}-2x^n+1)Q(x)^2$. I was not sure how to proceed from here. Should Luke's lightsaber have given away his plan? During Luke's confrontation of Kylo Ren toward the end of The Last Jedi we see that       the lightsaber Luke uses a blue lightsaber (which I thought was his original one but that really doesn't matter), not the green one of his own construction.   Given that Kylo Ren  and Rey has just     fought over and broken the blue lightsaber   Why did Kylo Ren not notice this "inconsistency" and realize that      Luke's physical body was not there? I have a problem with verb structure In this sentence:     Four items are responsible for three-fourths of the overall spending around the world, the most prominent share being that of food.   Why after the comma, the verb "being" is used? Why it does not use "is"? 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. Half exact functor which is neither right exact nor left exact Write down the sum of sum of  sum of digits of $4444^{4444}$ Let $A = 4444^{4444}$;  Then sum of digits of $A = B$; Then sum of digits of $B = C$; Then sum of digits of $C = D$;  Find $D$.  What should be the approach here? Convert XML/HTML Entities into Unicode String in Python Let $f(x)$ be a monotone non-increasing function such that $\int_{0}^{\infty}f(x) dx$ converges. Prove: $\lim\limits_{x \to \infty} xf(x)=0$. My question is, why can't I simply contradict any other possibility by using the Integral Limit Comparison Test with $1\over x$? I am, after all, to show that $f(x)=o(x)$ as $x\to \infty$. I really don't understand why monotony is crucial here. I could use some help. How to get the CPU cycle count in x86_64 from C++? It says that Cognito Forms can send email notifications when a form is submitted. Where would I find those email notification settings?   I would like for my website clients to be the recipients of those email notifications. Do they have to have an account or a verified email address?   Also, is the form data included in the email notification? Also, is there an advantage to using the WordPress Plugin vs the regular embed code? I am trying to display certain features in a shapefile-based layer using QGIS. I have found the subset function in the properties panel, which does exactly what i want, but is too tedious given that I have to manually sift through a large number of features. Is there a quicker way to do this?  I am using both shapefiles and postGIS best,  dietmar ODBC connection to Access database in 64-bit Windows Why is using a wild card with a Java import statement bad? When speaking with my mother a couple of days ago, I read to her a message I was sending to my cousin on her behalf ending with: "... the birthday of your youngest." [implying her child]  She immediately leapt on this and said that as my cousin only had two children, the use of the superlative was not permitted and the comparative must be used instead: "... the birthday of your younger child."  She was adamant that this was a solid grammatical rule that she was taught throughout her education.  As a younger Briton, I have relatively little formal grammatical education to older people who were better taught in this regard, so I usually defer to her on grammatical rule knowledge.  Is she right?  Are superlatives not permitted when the domain of the object is only two?  To me it seems bizarre.  The minimum or maximum of any set does not only exist when there are different minima and maxima, or indeed something which is neither. How to download a file over HTTP? I have a small utility that I use to download an MP3 file from a website on a schedule and then builds/updates a podcast XML file which I've added to iTunes.  The text processing that creates/updates the XML file is written in Python. However, I use wget inside a Windows .bat file to download the actual MP3 file. I would prefer to have the entire utility written in Python.  I struggled to find a way to actually download the file in Python, thus why I resorted to using wget.  So, how do I download the file using Python? Can you trip as part of a Tome of Battle Maneuver? Proof of $\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)$ I have seen this identity on Wolfram mathworld and in a comment to another similar trigonometric proof:    I can't seem to find a proof to this identity online anywhere. What method is used to prove this? In bash, how do I escape an exclamation mark? I want to do something like bzr commit -m "It works!". I can sort of escape the exclamation mark by doing bzr commit -m "It works\!". However, then my commit message includes the backslash. How do I escape the exclamation mark, while still ignoring the backslash?