Why doesn’t light escape a black hole?
Why is a black hole black?
"A friend of my father" vs. "a friend of my father's" — which one is right? Why? I have seen them both in my reference book, which is written by a native speaker. I wonder why there are two different phrases to describe the same meaning and am looking forward to some reasonable details to explain why this is so. How to get pixel data from a UIImage (Cocoa Touch) or CGImage (Core Graphics)? I'm studying for a midterm and need some help with proving summation     $$\sum\limits_{k=0}^n\binom{n}{k}2^k = 3^n$$   using the binomial theorem.  This is what I've been thinking so far:   In the binomial theorem, we set $x = 0$ and $y = 2$, so:  $3^n = (x+y)^n = \sum\limits_{k=0}^n\binom{n}{k}y^k$  $ = \sum\limits_{k=0}^n\binom{n}{k}2^k$  Am I getting this correct so far or completely wrong? Any help would be appreciated. Thanks. Why didn't Gimli know Moria had fallen? How does the Wizard spellbook work? How do I determine what spells are in my spellbook when creating a character? Do I simply have a spellbook with all Pathfinder spells? Or if I'm creating a level 6 wizard, do I start knowing all level 6 spells, and then have to learn the rest as I play the game? Is there a symbol for "unitless"? That's basically the question, is there a "right" way to implement operator&lt;&lt; ? Reading  I can see that something like:  friend bool operator&lt;&lt;(obj const&amp; lhs, obj const&amp; rhs);   is preferred to something like   ostream&amp; operator&lt;&lt;(obj const&amp; rhs);   But I can't quite see why should I use one or the other.  My personal case is:  friend ostream &amp; operator&lt;&lt;(ostream &amp;os, const Paragraph&amp; p) {     return os &lt;&lt; p.to_str(); }   But I could probably do:  ostream &amp; operator&lt;&lt;(ostream &amp;os) {     return os &lt;&lt; paragraph; }   What rationale should I base this decision on?  Note:   Paragraph::to_str = (return paragraph)    where paragraph's a string. How strongly can a 1mo be rocked? My 1 month old baby only seems to quiet down when I vigorously rock her (while I carefully hold her head). Can I cause her some (brain?) damage if I rock her too hard? Why do electrons not fully fill up all the orbitals sequentially? I am wanting to find the correlation between two variables: a categorical vegetation variable (with categories 1-4)  and a continuous variable of erosion (with values anywhere between 0.2 and 9). I think I would be required to use a non parametric test to find the correlation, but bit of a newby to stats so could anyone suggest the best test to use? I also wanted to know how sample size can affect these tests, ie with a sample size of 100 or 1000? How can I search in another language? Exporting video to .mov in Blender 2.79 In the past, Blender (at least the Mac version) has had the ability to encode video files as .mov, the format used by Quicktime Player. However, with Blender 2.79 this feature seems to have vanished, for both the Internal and Cycles render engines. The release log happens to have the entry "Video encoding settings have been simplified, along with the addition of a Constant Rate Factor (CRF) mode" which I suppose could be the cause.  Does anyone know what happened—whether it is the aformentioned reason or something else—and how to export videos as Quicktime .mov in Blender 2.79? How to overwrite only part of a file in c++ The essential gist of my question relates to what the real, unvarnished realities are of graduate, post-graduate, research, fellowship, and professorship attainment - when you aren't a single person with no pressing demands other than feeding yourself.   [EDIT: I am humbled by the incredibly helpful and thoughtful responses so far, all of which are +1 from me. I'll leave the question open a few more days before clicking 'accept' on any one, but you all have my gratitude and I hope others will benefit anywhere near as much as I have. ]  What I don't know is what succeeding in this path truly requires in time committment and family support.   Right now I can handle a 12-14 credit course schedule with 3.7-4.0 GPA (in the top 5-10% for college and program, dean's list, etc), work part-time, and still have time to be home and not have a burn-out/breakdown - and the academics are the easy part by far! I could do more, but I need to work more too.   What I really need to know is how sharply different is the time/dedication curve of graduate work and above. I realize each step is a filter that removes most of the entrants (most people don't get a degree, most who do don't attend/finish graduate school, most who do don't seek a PhD,  most who do don't seek/gain professorship...and on and on). But I don't know how severe the stepping is - I'm not talking "tenured professor at MIT or bust" - I realize that bar is many orders of magnitude higher.  Background Details: As an "adult, returning" undergraduate student with a (disabled) wife and kids, I'm halfway through a bachelor degree now. I've located a field of study (in a focus of Industrial Engineering) that has made me very interested in altering my life path to include seeking a Masters and maybe even a PhD/professorship (my father was a professor, so its not a purely novel concept to me). I've met and spoken with my advisor and 2 other professors with experience in related fields, and all is encouraging; I have a graduate school in mind, I've received lots of great advice on preparing for school (on this forum and in person), I test well (no GRE fears remain), good grades/class rank, good prospects for future strong referal letters, etc - a comparitively charmed existence for which I am very grateful! I just have to investigate whether I can make it through with an intact, cared for family!  Any guidance you could provide me, and other aspiring "non-traditional" students alike, would be appreciated immensely - thank you! I am taking an abstract algebra course and I am really interested about the topic.  So much so that I spend most of my time reading supplementary materials.  I consequently know a lot more theorems than the ones covered in class.    In a recent quiz, I used a theorem in one of my proofs that was not mentioned in the lecture since.  My professor gave me partial marks for my answer for the reason that we didn't cover this theorem in class even though the proof was completely valid!  I can't really see where he is coming from.  How would someone be able to use a theorem correctly if he doesn't know its proof?  I was quite baffled by his comment.   Do you agree with this?  EDIT: The professor just Emailed me and said that after thinking for a while, he decided to award me the full mark for the question. He also mentioned that he didn't want discourage me from studying the subject (since he saw most of my previous quiz grades were full marks) and I was passionate about it, but kindly requested that I include any proof for a theorem that I cite from now on in his exams. He also sent a broadcast message to the whole class indicating this. Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$ I'm repeating material for test and I came across the example that I can not do. How to calculate this sum: $\displaystyle\sum_{k=0}^{n}{2n\choose 2k}$? Is cross-posting a question on multiple Stack Exchange sites permitted if the question is on-topic for each site? It is possible to migrate a question from one Stack Exchange site to another by closing, but if I have a question that I think is on-topic for multiple Stack Exchange sites, is it OK to post it on both (multipost)?   For example, I have a question that's earned me the tumbleweed badge on SO and I'm not sure what the best thing to do with it is. It's about a web server so it might be answerable on Server Fault but it's really more of a programming thing, hence the posting the question on Stack Overflow.    Is there any way to make the question visible on multiple sites (crosspost) and then accept the answer wherever it came from? Dynamically Create a generic type for template I'm programming WCF using the ChannelFactory which expects a type in order to call the CreateChannel method. For example:   IProxy proxy = ChannelFactory&lt;IProxy&gt;.CreateChannel(...);   In my case I'm doing routing so I don't know what type my channel factory will be using. I can parse a message header to determine the type but I hit a brick wall there because even if I have an instance of Type I can't pass that where ChannelFactory expects a generic type.   Another way of restating this problem in very simple terms would be that I'm attempting to do something like this:  string listtype = Console.ReadLine(); // say "System.Int32" Type t = Type.GetType( listtype); List&lt;t&gt; myIntegers = new List&lt;&gt;(); // does not compile, expects a "type" List&lt;typeof(t)&gt; myIntegers = new List&lt;typeof(t)&gt;(); // interesting - type must resolve at compile time?   Is there an approach to this I can leverage within C#? Links to the Wayback Machine are broken in chat transcript is an example of the bug. It looks like this in regular chat:  That link points to: http://web.archive.org/web/20080409050311/http://personalpages.manchester.ac.uk/staff/daryn.lehoux/Tropes.pdf  In the transcript, it looks like this:  That link points to: http://web.archive.org/web/20080409050311/%3Ca%20href=  ... which doesn't work so well! Could this be fixed? I'm not sure if this impacts links to elsewhere. How to Implement Page Object and Page Factory Pattern in Selenium Webdriver?