Suppose that $c_0, c_1, \cdots , c_n$ are real numbers and that $c_0 + \frac {c_1}{2} + \frac {c_2}{3} + \cdots + \frac {c_{n-1}}{n} + \frac {c_n}{n+1} = 0$. Prove that the equation: $c_0 + (c_1)x + \cdots + (c_{n-1})x^{n-1} + (c_n)x^n = 0$ has at least one real solution between 0 and 1.
Problem: If   $$C_0+\frac{C_1}{2}+\cdots + \frac{C_{n-1}}{n}+\frac{C_n}{n+1} =0,$$  where $C_0,...,C_n$ are real constants, prove that the equation  $$C_0+C_1x+\cdots +C_{n-1}x^{n-1}+C_nx^n=0$$  has at least one real root between $0$ and $1$.  Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 4.
There were so many discussions about an Ubuntu with RR. They were so many pros and contras. At the end ubuntu wont be changed, the 6 month cycle system wont be changed. Thats fine!   But because there are so many people who would like to use an ubuntu with RR Im asking myself why isnt there an Ubuntu Derivate with RR. I couldnt found one, there are a loooot of Ubuntu Derivates but no RR. Is it just impossible? I mean Ubuntu = 6 months cycle, so the derivate have to be a 6 month cycle too? Is there no other option? Or are there already some smart guys trying to solve this problem? Thank you! How to install program to all user accounts in Windows? I have several accounts on my home PC. I want to install a program for all user accounts, not just the one I used to perform the installation. How do I go about doing this?  I installed a program from an admin account, but it doesn't show up in the other local accounts.  I was not able to find any documentation on how to do this.  I have Windows 8.1, if that matters. Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$, $\forall x\in G$. Prove that $H$ is a normal subgroup of $G$.    I have tried to using the definition but failed. Can someone help me please. How do I round a number in JavaScript? While working on a project, I came across a JS-script created by a former employee that basically creates a report in the form of  Name : Value Name2 : Value2   etc.  The peoblem is that the values can sometimes be floats (with different precision), integers, or even in the form 2.20011E+17. What I want to output are pure integers. I don't know a lot of JavaScript, though.  How would I go about writing a method that takes these sometimes-floats and makes them integers? Children’s books, one in space setting, the other in a jungle When I was little I remember getting these two books from the library.  Both books were paperback and flimsy.  The pages were wide.  Each book had full page illustrations with puzzles and mazes in them, but it was like you were immersed in this setting completely.  They were also like choose-your-own adventure books, but only in the way that you go through the book in a different order based on your results in the puzzles/games.    For example, there is a bridge to cross, or you can take this path.  If you choose to cross the bridge you go to page x.  On page x you have to pretend to cross the bridge but some planks are broken.  You can't skip more than a certain number of planks, but you also cannot "step" on a broken plank.  One book was in a jungle setting with rope bridges, alligators, quick sand, etc. The other was in space, with all kinds of aliens and weird stuff.  I have faint memories of these books but I REALLY want to track them down because I remember loving them. Drawing text on an image background that has dark and bright parts Often, I want to write text over a background image. If the background image is bright, I use a dark color for the text, and vice versa. But, often the image has both dark and bright parts. So a dark text is unreadable on the dark parts, and a bright text is unreadable on the bright parts.  I currently use one of several solutions, all of them are imperfect:   Brighten the image and use a dark font, or darken the image and use a white font; the problem is that it reduces the beauty of the background image. Use grey text (as in this question:  ). But grey text often looks unappealing, and is still only barely readable. Use dark text, and try to position it on the bright spots of the image (or vice versa). This takes a lot of time because it requires me to position each word separately.   Are there better solutions to this problem?   In particular: is there an automatic tool that takes an image and a text, and positions the text only on the dark/bright parts of the image? How to sum a variable by group Explicit bijection between $\Bbb R$ and permutations of $\Bbb N$ PGFplots foreach equivalent to TikZ's with multiple variables separated by a slash Are you allowed to use numbers as table names in MySQL? I'm thinking of having a program that dynamically creates new tables as the need arises. Can I have tables named with just numbers in MySQL? What requests do browsers' "F5" and "Ctrl + F5" refreshes generate? Is there a standard for what actions F5 and Ctrl + F5 trigger in web browsers?  I once did experiment in IE6 and Firefox 2.x. The F5  refresh would trigger a HTTP request sent to the server with an If-Modified-Since header, while Ctrl + F5 would not have such a header. In my understanding, F5  will try to utilize cached content as much as possible, while Ctrl + F5  is intended to abandon all cached content and just retrieve all content from the servers again.   But today, I noticed that in some of the latest browsers (Chrome, IE8) it doesn't work in this way anymore. Both F5 and Ctrl + F5 send the If-Modified-Since header.   So how is this supposed to work, or (if there is no standard) how do the major browsers differ in how they implement these refresh features? Limit Decimal Places in Android EditText How is this command legal? "> file1 < file2 cat" Assuming file2 already exists, the command   &gt; file1 &lt; file2 cat   appears to be copying the content of file2 to file1.  But I cannot understand this structure.  I understand that "Nothing" is being directed to file1 (creating or erasing its content). Then the content of file2 is being directed to file1.  Why is cat after file2? How does it know to cat file2 if the operands are not in the correct order? I want to display the pages 'famous' and 'mschief'.  Using this code, WP_query() doesn't return any result:  $args = array( 'post_type' =&gt; 'page', 'pagename' =&gt; 'famous,mschief', 'posts_per_page' =&gt; 3 );   This code returns one of the 2 pages:  $args = array( 'post_type' =&gt; 'page', 'pagename=famous,mschief', );   And this code returns the 2 pages correctly:  $args = array( 'post_type' =&gt; 'page', 'pagename=famous,mschief', 'posts_per_page' =&gt; 3 );   Why does the first piece of code (the only correct one to me) doesn't work? And why the two others produce these results? How do I tokenize a string in C++? Java has a convenient split method:  String str = "The quick brown fox"; String[] results = str.split(" ");   Is there an easy way to do this in C++? Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$? A prelim problem asked to prove that if $X$ is a compact metric space, and $f:X \to X$ is an isometry (distance-preserving map) then $f$ is surjective. The official proof given used sequences/convergent subsequences and didn't appeal to my intuition. When I saw the problem, my immediate instinct was that an isometry should be "volume-preserving" as well, so the volume of $f(X)$ should be equal to the volume of $X$, which should mean surjectivity if $X$ is compact. The notion of "volume" I came up with was the minimum number of $\epsilon$-balls needed to cover $X$ for given $\epsilon &gt; 0$. If $f$ were not surjective, then because $f$ is continuous this means there must be a point $y \in X$ and $\delta &gt; 0$ so that the ball $B_\delta(y)$ is disjoint from $f(X)$. I wanted to choose $\epsilon$ in terms of $\delta$ and use the fact that an isometry carries $\epsilon$-balls to $\epsilon$-balls, and show that given a minimum-size cover of $X$ with $\epsilon$ balls, that a cover of $X$ with $\epsilon$-balls could be found with one fewer ball if $f(X) \cap B_\delta(y) = \emptyset$, giving a contradiction. Can someone see a way to make this intuition work? Location of /etc/fstab in El Capitan I want to activate the . But I can't find the /etc/fstab on my mac. osx 10.11.5 El Capitan I often see $I(t)\propto |E(t)|^2$. What is the exact form ? Which constants are missing to make this an equality and why are they so often omitted ? How can I select a point without selecting its reflected side Can Minecraft be played by two players locally on the same account? I just bought Minecraft. Now, my brother wants to play with me but my Mom doesn't want to buy another. We use the same Wifi network. Can we do this?