let $x_1, x_2, ... , x_{2014} $ be the roots for $x^{2014} + x^{2013} + ... +x +1 = 0 $ find $\sum_{k=1}^{2014} 1/(1-x_k) $ let $x_1, x_2, ... , x_{2014} $ be the roots for $x^{2014} + x^{2013} + ... +x +1 = 0 $ find $\sum_{k=1}^{2014} 1/(1-x_k) $.  So What I tried to do is to consider x^3 function and get the elementary symmetric functions. the find the sum of  $\sum_{k=1}^{3} 1/(1-x_k) $. I lost and did not work.  How I can approach this problem?
If $x_1,x_2,\ldots,x_n$ are the roots for $1+x+x^2+\ldots+x^n=0$, find the value of $\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$ Let $x_1,x_2,\ldots,x_n$ be the roots for $1+x+x^2+\ldots+x^n=0$. Find the value of   $$P(1)=\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$$   Source: IME entrance exam (Military Institute of Engineering, Brazil), date not provided (possibly from the 1950s)  My attempt: Developing expression $P(1)$, replacing the 1 by $x$, follows $$P(x)=\frac{(x_2-x)\cdots (x_n-x)+\ldots+(x_1-x)\cdots (x_{n-1}-x)}{(x_1-x)(x_2-x)\cdots (x_n-x)}$$    As $x_1,x_2,\ldots,x_n$ are the roots, it must be true that   $$Q(x)=(x-x_1)\cdots(x-x_n)=1+x+x^2+\ldots+x^n$$ and  $$Q(1)=(1-x_1)\cdots(1-x_n)=n+1$$ Therefore the denominator of $P(1)$ is  $$(-1)^{n} (n+1).$$ But I could not find a way to simplify the numerator.  Another fact that is probably useful is that  $$1+x^{n+1}=(1-x)(x^n+x^{n-1}+\ldots+x+1)$$ with roots that are 1 in addition of the given roots $x_1,x_2,\ldots,x_n$ for the original equation, that is  $$x_k=\text{cis}(\frac{2k\pi}{n+1}),\ \ k=1,\ldots,n.$$  This is as far as I could go...    Hints and full answers are welcomed.
Disable the activation of the menu bar when Alt is pressed in Windows 7 In Windows 7 how can I disable the function that activates the menu bar when Alt is pressed?  Are there some registry values to modify this behaviour? Is uncertainty principle a technical difficulty in measurement? Is the uncertainty principle a technical difficulty in measurement or is it an intrinsic concept in quantum mechanics irrelevant of any measurement?  Everyone knows the thought experiment of measuring the position of an electron. One can detect electron's position by hitting it with a photon, due to Compton scattering the collision of the photon with electron will change electron's momentum. This experiment is used to explain uncertainty principle to layman, but it is over simplified, isn't it? It also gives an impression that if there was a better suited experimental method the uncertainty principle becomes irrelevant.   I personally think it is intrinsic as it arises from the non-zero commutator of position and momentum operators irrespective of the measurement process. Am I right?  EDIT: My question is similar to certain extent to  and . The answers there are nice but they focus on explaining basics of quantum mechanics more than they comment on the technical difficulty part. In the answers of question 2 there are statements like "So, it's not a knowledge limit" and "you're sort of correct when you say it's an observational limit" without further comments  To summarize, assume hypothetically we managed to find a way in the future where we can have a look at an electron without disturbing it by measurement or causing its wave function to collapse, would the uncertainty principle still hold in such a case?? Why/Why not? Currentsection TOC in scrartcl? I like to get a toc at beginning of some of my sections, like in beamer where I can use \tableofcontents[currentsection] is there a possibility in scrartcl?  Something like this:  \documentclass[10pt,a4paper]{scrartcl} \usepackage[ngerman]{babel}  \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document}  \tableofcontents %complete  \section{First}  \section{Second} \tableofcontents %just subsection and subsubsection \subsection{SecondSub} \subsubsection{please} \subsection{this} \subsection{SecondSub2} \subsection{parts}  \section{Third} \subsection{another}  \end{document}   Looks like:  Inhaltsverzeichnis %complete  1 First  2 Second 2.1 SecondSub.......................2 2.1.1 please........................2 2.1.2 this..........................2 2.2. SecondSub2.....................3 2.2.1 parts.........................4  3 Third 3.1 another ........................5   1 First  2 Second  Inhaltsverzeichnis %just subsection and subsubsection 2.1 SecondSub.......................2  2.1.1 please........................2  2.1.2 this..........................2 2.2. SecondSub2.....................3  2.2.1 parts.........................4  2.1 SecondSub  2.1.1 please   2.1.2 this  2.2. SecondSub2  2.2.1 parts   3 Third  3.1 another What’s the rule for adding “-er” vs. “-or” when forming an agent noun from a verb? What’s the rule to decide whether you add -er or whether you add -or when creating an agent noun from a verb?    Sometimes it’s -er:      read > reader hate > hater hit > hitter   But other times it’s -or:    meditate > meditator collect > collector Why are some galaxies flat? What is the explanation for the flatness of some galaxies?   (If it's due to their rotation then why they are rotating, why some other galaxies are not flat etc., I would like to hear a nice and complete answer :) ) Is there a nice solution to get a "presenter mode" for Latex presentations? I really like the way Keynote and MS Office (and also OpenOffice &amp; similar) support a presentation mode that displays the current slide, the next (and maybe previous) slide, the elapsed and current time, and also any notes attached to the slides on one screen and the presentation itself on the beamer/second screen.  It enables me to give good, professional presentations without having to know everything by heart (I suck at that).  I found a , but I am looking for alternatives, especially OS-independent would be great. Not having to create 3 different PDFs is a boon.  Any hints? Is the CMB rest frame special? Where does it come from? It seems that we are . This is the speed of our galaxy relative to the cosmic microwave background. Where does this rest frame come from? Is this special in any way (i.e., an absolute frame)? EDIT: I think the more important question is &quot;where does the CMB rest frame come from?&quot;. Turn off paragraph indention for a single line Is it possible to turn off the paragraph indention for a single line? I have the following use case: I open a new section which describes the task of an exercise. Since I use the font size \large for the section I don't want to use the section text for the task description.  Therefore I added the task description after the section. The disadvantage is that an indention is created when starting to write text after the task description.  What are possible solutions to this problem? Upgrade SQL Server 2000 database to 2008 R2 and enable new features I have recently upgraded SQL Server 2000 database to 2008 R2.    What I did was:   Shutdown SQL Server 2000 (express) service on old machine, Move datafiles (mydatabase.mdf and mydatabase.ldf) to new machine, Run SQL Server Management Studio 2008, Connect to local database engine, Attach datafiles to database. Change the compatibility level of database to SQL 2008 (100).     Question: What else should I do to make migration complete?  I want to:   use new features like checksumming and full recovery model, make this database to be exactly as it was created in SQL 2008 R2, make this database to be fully compatibile, correct and be perfect suited for new, SQL 2008 R2 database engine.     In other words: I just want to know how to correctly and completly convert old SQL 2000 database to new 2008 R2 database, be calm that everything is done right and be happy with all new features.    I'm asking this question, because I've found a lot of sites on the Internet that says so many different things that makes me confused: some say that it is required to rebuild indexes, another says to do other things... and now I know nothing so I want to hear experienced person opinion and clear, step-by-step instructions. I work for very small company, I'm on my own and I don't want to screw things up.    Sir, I'm really impressed with your answer, I wasn't expecting so much.    So some comments:   The database is now in production. As I said, it was upgraded using deattach-attach method as I desribed in first post and as described on MSDN:  It had to be done quickly, so I was forced to do it that way. Let's forget about how inappropriate could it was and focus on current situation. The users/persmission is not a problem here - there are only few and permissions are simple. The application which use database is compatibile with SQL 2000 till 2012 so this is not a problem either. The database file (MDF) isn't big - only about 1GB.     Few more questions:   You recommend to use backup/restore method, but I did as written above, so can I encounter any problems now? Everything worked without any problem. About checksumming and full recovery model: it wasn't available/enabled on SQL 2000, so I want to use them now. You said that the only thing I need to do is to enable those options in database properties? I've read somewhere, that it isn't enough and I should also rebuild indexes or something. I really don't know, I just asking. I'm prepairing to migrate this database to SQL 2012 - so first it was from SQL 2000 to 2008 R2, now it will be from 2008 R2 to 2012 (it was impossible to do this directly because of lack of support of SQL 2000 databases in SQL 2012). So I understand that I should follow your guide: backup it in 2008 R2 and restore in 2012, then do the rest of your tips, right? Please explain me the backup/restore method: Is it like a dump of database to SQL queries and then restoring it by executing a bunch of queries? Will this method by the way "defragment" my database? If not, how to defragment/optimize it manually? As we were using SQL 2000 Express for years (no management interface), we were doing backups simply by stopping engine and RAR the DATA directory. For now, as we are on SQL 2008, isn't this still better than using backup function in Management Studio? Full recovery mode with frequent Transaction log backups - Where is the Transaction log stored - is it the LDF file? How sould I backup it properly?     I know that my questions may sound silly, I'm not professional database admin, but I am the only person here who can do such "hard core" task like upgrading the database engine. I'm also sure that your knowleadge will help a lot to other people like me.  Thank you very much for your time and knowleadge, I really appreciate this. Can I find out which tags I have created? How to recover a removed file under Linux? By accident, I used rm on a file I didn't want to delete. Is there a way that I can get it back under Linux? In Java when one interface extends another, why would one redeclare a method in a subinterface? I've been looking at the JMS API from J2EE and found a strange behavior where certain methods that are declared in an interface (e.g., createQueue in Session) are declared again in the subinterfaces such as QueueSession, and with identical documentation.  Since an subinterface "inherits" all the method declarations of the interface it inherits, and since the JavaDoc tool has no problem sorting out the JavaDocs of the subinterface and creating an "inherited operations" list, I can't figure out what this achieves.  The only think of is that initially the call was in Session, and then moved to QueueSession when the specific subclass was created, though then I would have expected to see something in the documentation of the upperclass. But this is just conjecture.  So the question is: Is there a convincing reason to redeclare a method in a subinterface? Require reason for post deletion and un-deletion? Could we have a method of explaining why we think a question or answer should be deleted?  It could work identically to closing a question, maybe with an additional text-input box (like the "Requires Moderator attention" flag option) to provide additional info because the reasons for deletion should really be more specific..  For example, "exact duplicate" doesn't explain to me the user has re-posted the same question, thus, it should definitely be deleted (not just left closed). Like with close-messages, these reasons would be listed under the deleted item (or answer).   I don't think comments are suitable, as they can be a pain to quickly.. "extract" deletion reasons from them - I have to scan through the comments, and they may all be irrelevant.  Equally, for un-deletion, it'd be useful to know why another user thinks a question should be restored. For example "new user deleted the question after they accepted an answer", or "Jeff went insane and started deleting any questions with the letter O in it".  I think requiring un-deletion reasons is possibly more important, since commenting is disabled on deleted questions - we have no way of explaining why we think it should be restored I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences  f(0) = g(0) = 1 f(n) = f(n-1) + 2g(n-1) g(n) = f(n) + g(n-1)   where $f(n)$ is the actual answer and $g(n)$ is a helper function that represents the number of ways to tile a $3 \times 2n$ rectangle with two extra squares on the end (the same as a $3 \times 2n+1$ rectangle missing one square).  By combining these and doing some algebra, I was able to reduce this to  f(n) = 4f(n-1) - f(n-2)   which shows up as sequence , confirming that this is the correct recurrence.  The number of ways to tile a $2 \times n$ rectangle is the Fibonacci numbers because every rectangle ends with either a verticle domino or two horizontal ones, which gives the exact recurrence that Fibonacci numbers do. My question is, is there a similar simple explanation for this recurrence for tiling a $3 \times 2n$ rectangle? Add different materials to different parts of a mesh? How do you add more than one material to a mesh?   I have tried adding two materials to the object, and then assigning one of the materials to a vertex-group, but I could not find a place to assign it.  I am making a striped bouncy ball that I want to deform, so adding a different material by separating the mesh into different objects would not be ideal. Has a free neutron ever been shown to absorb/emit/interact with a photon? Protons only 'interact with' very high-energy photons, whether inside a nucleus or free, right?  I'm assuming the same about neutrons....  Neutrons have a small magnetic moment and a slight electric 'moment' (dipole?), correct?  They are sometimes considered to consist of a negative charge surrounding a positive charge?  I would assume that neutrons, in a nucleus or free, interact with photons even less often than protons, and that the gamma photon has to have an even higher energy than one interacting with a proton....  Lastly, I have read about 'photodisintegration', in which a nucleon is kicked out of a nucleus by a gamma ray.  It seems to happen to neutrons sometimes...  But, I am mostly curious about experimental evidence of free neutrons interacting with photons, how often it happens compared to free protons, and whether the interacting photon's energy has to be extremely high, higher even than a photon interacting with a proton... Which book (free or otherwise) was the most useful to you when you started learning LaTeX?  I am frequently asked this question by friends who want to learn LaTeX, and I recommend the book which got me started, , but I feel that there might be better options around.  Also see . How many bandits must agree? Problem  A group of $n~(1 \leq n \leq 30)$ bandits hid their stolen treasure in a room.  The treasure needs to be locked away.  The bandits want to ensure that at least $k~(1 \leq k \leq n)$ of the bandits must agree in order to retrieve the treasure.  They have decided to place multiple locks on the door such that the door can be opened if and only if all the locks are opened.  Each lock may have up to n keys, distributed to a subset of the bandits.  A group of bandits can open a particular lock if and only if someone in the group has a key to that lock.  Given $n$ and $k$, how many locks are needed such that if the keys to the locks are distributed to the bandits properly, then every group of bandits of at least size $n$ can open all the locks, and no smaller group of bandits can open all the locks?  For example, if $n=3$ and $k=2$, only 3 locks are needed - keys to lock 1 can be given to bandits 1 and 2, keys to lock 2 can be given to bandits 1 and 3, and keys to lock 3 can be given to bandits 2 and 3.  No single bandit can open all the locks, but any group of 2 bandits can open all the locks.  My thoughts  At first glance, I thought it was just a simple combinations problem ($nCr$), and the provided test cases confirmed my suspicions: when $n=3$ and $k=2$, $\binom{3}{2}=3$.  Likewise, when $n=5$ and $k=3$, $\binom{5}{3}=10$.  However, two cases where this doesn't work are where $n=2$ and $k=1$ and $n=5$, $k=4$ (it gives $2$ and $5$, respectively, when the correct answer according to the judging data should be $1$ and $10$).  Next, I bothered a colleague who's a math major, and he briefly glanced at it and suggested it might be a hypergeometric distribution problem.  However, if this is correct, I'm having trouble wrapping my head around it - all the examples I can find have 2 distinct categories, e.g. red and green marbles being drawn from an urn, and I don't see that here.  An example in the terms of this problem would be extremely helpful.  If this isn't correct, where do I go from here?  I've been spinning my wheels on it for quite a while.  Background  This is for homework for a CS class (I'm a CS major with 2 semesters of calc plus discrete).  I'd like to be able to understand the math behind it, because once I can understand it, I can program it.  This problem originally came from an ACM ICPC regional contest this year, but my professor has made it into a challenge homework problem. What are the pros and cons of the leading Java HTML parsers? I know I use them, and my peers use them, but how much does the average user actually use tooltips to figure out functionality?