prove $\phi$ is linear
Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear?
This is a story that was read on BBC radio about thirty years ago, and this is about as right as I can remember it: The human pilot of a spacecraft is doing some reconnaissance on an alien planet. He crashes but survives. For some reason, no one will know where to look for him. He sees something in the distance that looks like a building of some sort. When he gets there, it seems to be a home. Very much like a human home. There's furniture, but it's strange and uncomfortable . There's food, but it's inedible. There's a faucet but what comes out of it seems more like acid. Everything looks as though it should be right, but it's not. He leaves the house, wanders and after a few days is desperate for water. But there isn't any water anywhere. He's close to death, crawling on the ground and then something happens. There are cracks in the soil and there appears to be water bubbling up from perhaps some underground stream. The water revives him. He goes back to the house and tries the food again. It's delicious and he eats enough to gain back his strength. Then he decides to try the water again. It's pure and wonderful. He takes a shower and the last line of the story is how refreshing the water is on his skin and something about him lifting his tail. Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$ Can you be sued for using publicly used images on a blog without copyright permission? Relationship between SVD and PCA. How to use SVD to perform PCA? Principal component analysis (PCA) is usually explained via an eigen-decomposition of the covariance matrix. However, it can also be performed via singular value decomposition (SVD) of the data matrix $\mathbf X$. How does it work? What is the connection between these two approaches? What is the relationship between SVD and PCA?  Or in other words, how to use SVD of the data matrix to perform dimensionality reduction? Approximation of measurable function by simple functions I am currently trying to develop my modeling skills, and following a tutorial found on YouTube. The tutorial teaches how to create an anvil. At 19:10 or so into the video, a subdivision surface modifier is used to create a smooth, soft look.  However, when I do the same, mine looks very sharp and generally bad.  Any ideas why? I will include the blend file below, I appreciate your time and suggestions.    This is his when he subdivides, which is very different from mine when I do the same at the same settings. (I cannot post more pictures because I am new, but if you check the blend file I attached, you should be able to see the difference.   This is what it looks like when I subsurf without the base face. What is the relationship between the focal length and field of view for an equal-area ("equisolid") projection fisheye lens?  Alternatively, what is the relationship between the 180-degree image circle size and the focal length for such a lens?  Ultimately I want to find the field of view of a Tokina 10-17 mm fisheye zoom at focal lengths other than 10 mm on APS-C (). Is abstract algebra (mostly?) restricted to $2$-ary operators? This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic objects consisting of a set together with one or more binary operators defined on that set, perhaps over some other algebraic structure. Not counting categories here, just "low-level"-ish stuff.  I'm an undergraduate so my knowledge is of course very limited but I can't help to wonder why I never stumble upon algebraic structures with $n$-ary operators? Is there a good reason for this, something along the lines of $n$-ary operators behave "badly" when $n&gt;2$ or is it just my ignorance because I'm a beginner in the field? Is it possible to schedule my laptop to mute and unmute at a specific time? I always used to wake up to my favorite radio station on my alarm clock. However, I have moved and can no longer get that radio station over the air. To solve this problem, I would like to stream the station on my laptop, but have it muted overnight, then set it to unmute at a specific time in the morning, so that it will act as an alarm clock. Is there a way to schedule my laptop (Windows 7 64-bit) to unmute at a specific time? Why is defragmentation unnecessary? Why is defragmenting unnecessary in Ubuntu? Voting on Meta sites generally express agreement and disagreement with a proposal, unlike on the Q&amp;A sites where they indicate useful/good vs. non-useful/bad content.  It seems to me that a down-vote on Meta Stack Exchange should not carry a -1 rep penalty, which only discourages down-voting. With this discouragement, questions on Meta SE will be biased towards agreement, since for every up-voter, there may be several in disagreement who don't want to lose the rep for down-voting.  Can the penalty for down-voting on Meta SE be removed? I was looking for a tree or graph data structure in C# but I guess there isn't one provided.  explains a bit about why.  Is there a convenient library which is commonly used to provide this functionality?  Perhaps through a strategy pattern to solve the issues presented in the article.  I feel a bit silly implementing my own tree, just as I would implementing my own ArrayList.     I just want a generic tree which can be unbalanced.  Think of a directory tree.  C5 looks nifty, but their tree structures seem to be implemented as balanced red-black trees better suited to search than representing a hierarchy of nodes. Derive a formula to find the number of trailing zeroes in $n!$ Possible Duplicate:          I know that I have to find the number of factors of $5$'s, $25$'s, $125$'s etc. in order to do this. But how can you derive such a formula for any number $n$? Does the std::stack in the C++ STL expose any iterators of the underlying container or should I use that container directly? Quickest way to rename files, without retyping directory path Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical?  Let's assume that a planet is 'born' when lots of particles of uniform density are piled together, and constrained only by the Newtonian gravity of the $N$ particles with no external gravity.    My question is: Is it possible to use the variational calculus starting from this situation to prove that planets should be spherical? How does one change to engineering for graduate school after undergrad in a science field? I've always had an interest in electrical engineering (actually all engineering in general) and I've decided that I'd like to pursue a graduate degree.  I guess I'm wondering if I should really try to do this or just keep it as a fun hobby I spend all my free time doing.  So, the question is,  how does one change fields between undergrad and grad school?  A specific example follows, but please provide answers that address a more general case, so that your answers will be helpful to other people with the same general question.  I already have a BS and MS in chemistry with a minor in applied math.I published three papers during my MS, one first author in the top journal in my specialty and two as coauthor in an average and a so-so journal.  I hope those will count for something.  I finished my BS with around 3.1 and my MS with a 3.9 gpa.  I'm sure that my MS advisor would write me a great letter.  My GRE scores are too old to count so I'd have to retake it or try and get a pass on it since I already have a graduate degree.  Since finishing school I've been working for 4 years as a chemist.  I can get great recommendations from my supervisor who is a PhD chemist.      I've been working for a while to cover some of the things I would have learned in a BSEE to try and make up for my lack of one.  I worked through some texts on analog/digital design, lots and lots of programming, built a balancing robot that got a bit of internet attention, and I'm currently working on an autonomous robotics project that's looking really good so far.  After that I have more project ideas than I have time or money for.    I want to apply for PhD programs (specifically in controls/robotics) but I'm worried that my application will get tossed right away because I don't have a BSEE.  Maybe I should just apply to MS programs? Does Find My Mac work if the disk gets erased? I've just bought my first Mac, a Macbook pro(with an hdd), and I'm thinking in buying an ssd and replace the hdd. But the computer came with no dvd of the OS. I know that in the hdd there is a restore partition, but if I remove the hdd, how can I install the OS? Share menu hangs iOS app The Share menu in iOS 13 got a redesign, and the Stack Exchange iOS app is no longer compatible with it. I can still use it to copy the link, but when I try close the menu, it remains stuck. Opening another app and then returning automatically dismisses the menu, but then I can't use the back button anymore to return to the question list. (I'm still able to refresh the page, so it's a navigation issue.)    Ironically, I discovered this bug while trying to mark  as a duplicate; in the app, the quickest way to do that is to use the Share menu to copy the link to the original question to the clipboard...   App Version: 1.6.6.2 Device: iPhone 7 Plus OS Version: Version 13.1 (Build 17A844)