Evaluate $\sum_{k=1}^n\frac{1}{k}\binom{n}{k}$
Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $
I have lost my passport here in Beograd. Can I travel just with my Czech ID card from Belgrade to Budapest? How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$? I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property. Efficient way to check collision with potentially hundreds of objects I am trying to make a top down shooter that could potentially have hundreds of objects at once at one point. I need to check collision with projectiles, walls, and enemies. I am currently using ArrayLists which probably won't suffice later on and was wondering if someone had a good method or knows a good method for checking all of those objects at once. This is going to be a game for smartphones and is built in java. Any ideas would be great or if you could point me in some direction that would be fantastic.  Thanks in advance! Sum over multiple indices Path connectedness of the set $\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x>0\}$ Let $A$ be the following subset of $\mathbb{R}^{2}$:$$A=\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x&gt;0\}$$ Then  $1.$ $A$ is connected.  $2.$ $A$ is compact.  $3.$ $A$ is path connected.  $4.$ $A$ is bounded.  It is clear that $A$ is unbounded as the graph of $x\sin(\frac{1}{x})$, so option $2$nd and $4$th are not correct. Again $A$ is connected by using the fact that closure of connected set is connected. But i am confused about path connectedness of $A$. Please help me about path connectedness of $A.$ Thanking you. Legally possible to re-create/remake an old game? Recently I started remaking an old game on Unity. Basically I want to remake an old RPG that was released for Playstation 1. I'm not ripping anything from the old game, all the assets are being created by me. I'm creating everything from the scratch, music, 3D models, gameplay however I will keep the (copyrighted ?) names, I will try to create the characters and maps/levels as they were in the original version of Playstation 1 but with better graphics.  I'm not planning on selling it or adding in on a platform like Steam (for obvious reasons, copyrights etc), just giving it for free. But I wanna know when I will finish the development, can I share the game on public with other people so they can download it?  Can the company that made the original game sue me/fine me because of copyrights reasons even though I'm not planning on selling for money?  tl;dr Remake an non-profit game of PS1 from the scratch, can the company that holds the copyrights of the game sue me/fine me? How to find unused/dead code in java projects What tools do you use to find unused/dead code in large java projects? Our product has been in development for some years, and it is getting very hard to manually detect code that is no longer in use. We do however try to delete as much unused code as possible.  Suggestions for general strategies/techniques (other than specific tools) are also appreciated.  Edit: Note that we already use code coverage tools (Clover, IntelliJ), but these are of little help. Dead code still has unit tests, and shows up as covered. I guess an ideal tool would identify clusters of code which have very little other code depending on it, allowing for docues manual inspection. Can I "fill" an empty object? Picking coefficient in Inverse Distance Weighted method of QGIS? I'm using QGIS 1.8.0 to interpolate weather data. This is, I have a weather station located in say, A and C. With the Inverted Distance Weighting or Triangular Interpolation methods, I can know/interpolate information in B. When using the IDW method, I have to choose a coefficient.   Does anyone know in which range the coefficients should be?   Say from 1 to 10 or .1 to .99 or 100 to 1000?   Any ideas? AlwaysOn AG: Current Host Server - Owner Node - Primary Instance What does standard deviation tell us in non-normal distribution In a normal distribution, the  imparts standard deviation a lot of meaning, but what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data values still fall within 3 standard deviations? Do we have rules like the 68-95-99.7 one for non-normal distributions? Possible Duplicate:          The FAQ specifies that:     Use your downvotes whenever you encounter an egregiously sloppy, no-effort-expended post, or an answer that is clearly and perhaps dangerously incorrect.   But if I find a correct question, closed (or not yet) as duplicate, is downvote a correct behavior for Stack&nbsp;Overflow policy? Can anyone explain why a semi-simple finite dimensional Lie algebra $\mathfrak{g}$ has to be perfect ?   The natural way to prove something like that would be to look to the algebra generated by the Lie brackets, which when $\mathfrak{g}$ is not perfect would be expected to be solvable. But it doesn't seem to work. Enumerate alignment problem in theorem environment I'm using amsart. When I create an enumeration inside a theorem environment, the numbering is not aligned properly:  Theorem 14. (1) (2) (3) (4)   How do I shift the (2),(3),... to be aligned with (1)?  MWE:  \documentclass{amsart} \newtheorem{thm}{Theorem}[section]  \begin{document} \begin{thm} \begin{enumerate} \item Item 1 \item Item 2 \item Item 3 \end{enumerate} \end{thm} \end{document} PhD in Europe without scholarship from non-European homeland I earned my bachelors in Computer Engineering from my home country. I am now enrolled in a Physics masters program in the same university. Next year I plan to find a different university in Europe in which to enroll in a Computer Engineering PhD program.  However I doubt that I'll receive any scholarships from my homeland's government when studying abroad. Can I survive if I start to live out of my own my pocket? Is that enough to live in Europe just for a half year. Are there any available funds that can supplement my income such as a TA/RA's salary?  My main question is:   How to survive if I want to study PhD in Europe without any scholarship from my home country? Which country might provide sufficient support for a foreigner to study at graduate level?   I am interested to Germany and Finland but any European country would be acceptable! The Royal Order of Adverbs I know that the pattern manner-place-time shouldn't be taken too seriously if one wants to speak natural English.   In real life, people rarely use a string of adverbs. Speakers will easily break the pattern for emphatic or stylistic reasons (usually placing the adverb at the beginning of the sentence). Other factors such as length and specificity will also override the arrangement.  But I'm constructing illustrative sentences that highlight (and stick to) the Royal Order of Adverbs.   What all the online sources are doing is mixing adverbs and prepositional phrases, but I can't have the concept of prepositional phrases in my examples just yet. This means I should include only one-word or two-word adverbs (without prepositions).   So, I have these:    He sat quietly upstairs all day. She waited impatiently outside this morning. They gather noisily downtown every Saturday night.   Are the sentences above still correct, as in natural-sounding? Somehow they started sounding a bit off to me. I guess I just need to test them with a second pair of ears. What exactly happened with Dr. Crusher's husband and Captain Picard? scanf not taking in data Webforms and jQuery, how to match the ID's? I want to use jQuery with asp.net webfoms.  Do I need to get a special toolkit so the .net controls spit out friendly Control ID's?  Reason being, I don't want to write javascript referencing my html ID's like control_123_asdfcontrol_234.  Has this been addressed in version 3.5?  (I remember reading you have to get some special dll that makes ID's friendly). Is there a device that could measure the speed of time? Is there (or can there be) a device that could measure the speed and acceleration of time?