Over the reals, I know this is true: $$(x^r)^s=(x^s)^r$$ for $x&gt;0$ and $r,s\in\mathbb{Q}$.  Does the same equation hold over $\mathbb{C}$? I.e., $$(z^r)^s=(z^s)^r,$$ for $z\in\mathbb{C}-\{0\}$ and $r,s\in\mathbb{Q}$?
Regarding ,  it is not possible to combine complex powers in the usual way:  $$ (x^y)^z = x^{yz} $$  There was mention of multi-valued functions, is there some theory that makes this all work out correctly?
Why does the image not appear? Is the Moonbeam spell amazing, or are we doing it wrong? The description of the moonbeam spell (PHB pg. 261) says:  When a creature enters the spell's area for the first time on a turn or starts its turn there, it is engulfed in ghostly flames that cause searing pain... On each of your turns after you cast this spell, you can use an action to move the beam 60 feet in any direction.  We read this to mean a creature takes damage from moonbeam when any of the following conditions are met:  It begins its turn inside the moonbeam It enters the moonbeam on its turn The creature finds itself inside the beam on another character's turn, no matter how it got there  The creature will not take damage in the following instances:  A creature moves from one part of the beam to another (obviously it had to take damage, but it won't take more damage) A creature is in the beam after having been inside the beam already on that turn.  Following these assumptions, our druid cast a moonbeam and did damage to two enemies, Bob and Frank. On their turn, they took damage again, killing Bob. Frank left the beam. On the druid's next turn, she moved the beam in a circular motion, not exceeding 60 total feet of movement, and in the process the beam came in contact with a dozen enemies and ended on Frank. All dozen enemies received damage, as they were inside the beam at some point during the turn. On Frank's next turn, he again took damage for the fourth time and died. On each of the druid's turns, she moved the moonbeam around like this until the fight ended (it did not exceed the time limit and she made all concentration checks). Is this a legitimate use of moonbeam? It seems awfully powerful for a level 2 spell. Why is it bad to log in as root? I've often come across posts on forums or other websites where you see people joking in such a manner about running/logging in as root as if it's something awful and everyone ought to know about it. However, there isn't much that a search reveals on the matter.   It may be widely known to Linux experts, but I really don't know why. I remember always running as root when I first tried Linux years ago (Redhat and Mandrake) and don't remember running into any problems because of that.   There are actually some distros that have a bright red background with alert signs all over it as wallpaper for the root user (SuSe?). I still use the "Administrator" account for regular use on my Windows installation and haven't ever run into any problems there either. Any command that starts with $ returns  command not found   How can I fix this?  I am  using 16.10 Ubuntu. Can anyone help me identify a story about kids with silver blood? What is the preferred way to mention Stack Overflow from Twitter? I wanted to tweet something on the lines of "Stack&nbsp;Overflow is awesome". When I do this kind of stuff, I usually use a username (for example: ) or a hashtag ()  I found , but it seems it is from someone unrelated with Stack&nbsp;Overflow.  Is there a Twitter account I can reference, or should I use a hash-tag? Should I just point to the site?  Thanks!  (Notice: I've seen lots of questions regarding Twitter and Stack&nbsp;Overflow, but those are mostly about integration/login with Twitter for the users accounts; I'm just asking for a single account for Stack&nbsp;Overflow; probably for announcements and PR, which is different) What happens if both players die simultaneously? If both players are very low health and someone plays this card:    Both players could die simultaneously. What happens in Arena when this event occurs? Does it count as a loss for both players? Does it count at all for your Arena record?  Side note: Holy Nova + Auchenai Soulpriest could cause the same scenario. I never liked that SO/SE relied on the external gravatar service, so I was glad to . However, it turns out that one cannot use as profile picture a PNG file with transparent background (something that is supported by gravatar):    Above is a screen capture of my profile page, with a profile picture that has a transparent background. That background is turned into black, which I suppose is a bug.  The method of upload doesn't change anything (I uploaded it manually to imgur, but the same black background is observed). See the difference between the two following URLs:  http://i.stack.imgur.com/Da3ex.png?s=128&amp;g=1   and  http://i.stack.imgur.com/Da3ex.png   The problem is thus in the resizing code from imgur. What does 1x1 convolution mean in a neural network? I am currently doing the Udacity Deep Learning Tutorial. In Lesson 3, they talk about a 1x1 convolution. This 1x1 convolution is used in Google Inception Module. I'm having trouble understanding what is a 1x1 convolution.  I have also seen  by Yann Lecun.  Could someone kindly explain this to me? Is it possible to change your spawn point? I have a raster aerial photo.  I have county wide parcel data.  I want to clip/screen shot the raster at each individual parcel.  Think of this application - a post card with an image of your house --here is your house--  I want to have each parcel attribute to contain information + hyperlink of the aerial photo.  Any other solutions welcome, but this is the only method I could think of. Although I am unsure of how to achieve it efficiently.  Using ArcInfo 10. I am interested in theoretical and practical considerations. Summation by parts (finite calculus) of $\sum\limits_{i=0}^{n}i\cdot a^i$ I'm studying the calculus of finite differences and have read about summation by parts:  $\sum f(x)\Delta g(x) = f(x)g(x) - \sum g(x+h)\Delta f(x)$.  The tutorials I'm using go a bit woolly at the point of introducing this technique: they use examples where they set $\Delta g(x)$ to $2^x$, and don't really show how to use it for anything a bit harder. I'm at a loss for basic examples.  I wondered if it is possible to determine  $S.n = \sum\limits_{i=0}^{n}i\cdot a^i$  using summation by parts?  If so, how is it done?  Thanks in advance. Reference to a figure uses the section's number When I see someone else being embarrassed / humiliated, for example a singer forgetting their words in a concert, I would say something like "I feel embarrassed for them". But is there a better word to describe this feeling?  I would say it's a mix of:   Empathetic discomfort for someone else's humiliation Tension; hope that spectators will be forgiving Frustration that I cannot help them   Other scenarios I would feel the same emotion:   Seeing someone make a presentation, the slides are all out of order, and their face gets red with embarrassment while there's a tense silence in the room. Seeing someone make an embarrassing social blunder in front of unforgiving people. Seeing a speaker who keeps making a distracting and annoying gesture, and they don't realize that the whole audience is annoyed. Seeing someone be taken advantage of, like seeing someone naively and happily accept a terrible offer on a car. Seeing a person of strong image (for example the CEO of your company) do something like accidentally leave their webcam on after a conference call, and employees see him/her let out a big burp and pick their nose or something.   In these cases it's important that the audience is not forgiving, even if they don't confront the person about it. The person may or may not know they are being humiliated.  "Pity" as proposed in an answer seems very close, but I think it lacks a few components. I think "pity" has a chronic connotation - that you are referring to the general state of things rather than a specific event.  I think "pity" also depends on the subject suffering, but in my example they are often oblivious.  It could very well be that there is no word for this at all.  NOTE: Some people do not feel the same way I do in this scenario, so you may not think this question makes any sense. Wear a hat multiple times? Explain briefly What is meant by interpolation.How is it related to the concept of regression?  interpolation is art of reading between the lines of a table and in elementary mathematics the term usually denotes the process of computing the intermediate values of a function from a set of given or tabular values of that function.  I can't give the answer of second question. We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or to the indeterminate product $ 0 \cdot \infty $?  Furthermore, why can’t we call $ \infty + \infty $ and $ \infty \cdot \infty $ indeterminate forms?  I’m new to calculus, so please clear up my concepts if you can. Thanks! Story where vampires are created by drop absorbed on neck? I have been trying to identify a book that I read maybe 25-30 years ago (1987-1992), in which there are vampire or vampire-like creatures, but instead of being created in the traditional way (e.g. bite on the neck), something drops a drop of a fluid on the back of their neck, which is then absorbed into the body and transforms them. I have a vague notion that some sort of tree is what creates these drops. The drops may have had a shiny metallic appearance.  I've tried to find this book before and always came up short. I'm starting to think the whole thing was a dream and I never read it at all. But I really feel like it was a book or at least a short story. Distance learning: taken seriously? I'm currently in the first year of a six year bachelor's degree aiming for astrophysics. The course is a home/distance learning setup run by the Open University. While I am enjoying it thoroughly, I am aware of the academic politics that can dog professional science. So my question is thus: Upon soldiering on with the degree, then masters, and then PhD, will I be taken seriously within a professional setting given my inability to reference a leading university for my education? Will the format for my studying be "pitied"?   Edit: I suppose a bit more info may help. I'm 27,wife, 2 children, and I currently work as a full time chef, and self employed part time IT technician. I flunked my GCSEs, with no further education. The past 3 years have been a revelation for me, and I have discovered a before unexperienced love for academia; specifically, physics and astronomy. Last year I completed a 1 year OU maths brush up course, which at the time I really struggled with. This year I had a bout of madness and took the plunge for a BSc degree in astronomy. I am now loving the math within the physics so far (yet to hit calculus though!)  PhD, and even masters, is a long way off.  Also, since this original post I've improved my understanding of what a PhD actually entails. So I realise that if I get there, it may well not have anything to do with the OU!     It occurs to me that a lot of my initial queries could have been avoided with a better understanding of the educational and qualification system itself. I now have a greater appreciation as to what a PhD actually is, realising that its a long way off and not necessarily needed for a fulfilling career in research.  I think trying to repair my broken education and fight my way into a worthwhile (and productive!) career is enough of challenge at the moment. There will be plenty of time later for worries concerning doctorates.  It is good to hear that there is at least an open mind towards distance educated professionals.   As has been said, if you've got it, you've got it - you just have to be ready to prove it.