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Root of unity for $n+1$ Let $n \ge 1$ and $u_{1},u_{2},...,u_{n}$ are all root of unity for $n+1$ and there are different from $1$.(a) Find polynomial $f \in \mathbb R[x]$ of degree $n$ for which: $f(u_{k})=0$ for $k=1,2,...,n$.(b) Calculate real coefficient and imaginary part for $(2-u_{1})(2-u_{2})\cdot...\cdot(2-u_{n}).$    I have a knowledge about root of unity but I completely don't know how to do it because I think that I need some extra trick to do it but my ideas finished. I know that $\varepsilon_n^{(k)} = \cos\left(\tfrac{2k\pi}{n}\right) + i\sin\left(\tfrac{2k\pi}{n}\right) = e^\frac{2\pi i k}{n}$ and for $n&gt;1$ $\sum _{{k=0}}^{{n-1}}e^{{\frac  {2\pi ik}{n}}}=0$ but I think it is too little knowledge to solve this task.Can you help me? How can I get a pure white background in jewelry photography, without losing sparkle and shine? I've read through several threads similar to this, but I'm looking for a comparison of two options here.   I currently take photos on a raised glass platform against a white foam board background and run about 300w of lights (3 100x LEDs @ 5000k, Home Depot Specials in softboxes and umbrellas). As you can imagine, it takes me about 15 minutes of post processing to achieve a perfect white background, and sometimes I have a particular photo that I am unable to get there.  I want to upgrade to either a photography table with a translucent background that I can place a speedlight under, or a cyclorama infinity background such as the MyStudio MS20. The latter option may require an upgrade to my lighting, which I am planning to do anyways.  The idea of course is to get a blown out 255 background with no post processing, other than color correction on the jewelry or product as needed. This would cut my time after the shoot to a ridiculously small amount of time.   Which option would make my life easier?   For reference, I currently use a D5300 with a Tamron 90mm f2.8 macro lens.  This is the best shot I got after quite a lot of effort, before and after processing:      Note that the silver is not shiny like it should be, and the sparkle on the gems is lost. A recent quandary came up with some friends in a game of Ticket to Ride.  If, in the last turn, you have the cards to play a route longer than the number of trains that you have left, can you do so?  Or are you limited by your number of trains? Does an adjective or longer modifier apply to both nouns when they are joined with 'and' [or 'or']? Can you grab the blue shirts and socks?   Is the above sentence stating that both the shirts and the socks are blue? Or only the shirts?  At this stage, I am leaning towards the earlier (only the shirts) — though writing "Can you grab the blue shirts and blue socks?" seems redundant. bounding a $C^1$ function that upholds $f'(t)\leq 1-f(t)^2$ Construction of "it's too hard a task" I want to write as a math fraction 1/4 " ( a quarter of inch). I want to write this fraction out of an equation because is a characteristic of an element. I have gridded depth points in CSV format that I'm trying to add other map data to, in additional columns. I've tried joining fields, hoping that I could link the Shapefile data to the csv, but when I try to display the csv by graduated symbols based on the new fields, it goes blank. I used Nathan's tips from , but imported the csv as csv since I want to add map info to that, rather than adding csv info to a map.  See this picture below for the sea bottom temperature &amp; salinity voronois by month, underneath the csv points (a sample subset), with the joins already made. I added columns on the csv for each of the fields i'm intending to append.    I also tried spatial joins as per  however it says the layers aren't the same CRS (I checked, they are), then gets to 15% and hangs.  I also also tried spatial join through  but it hit a python error. I can try this again and look to debug the problem if this is the right way to go.  In short: appending polygon data as fields on a points-grid csv: what's the best way in QGIS 2.0? How to avoid the wrinkle at the poles of the UV sphere? How come such different methods result in the same number, $e$? So I've been experiencing this in the last few days: sometimes when someone replies to my comment, and I reply to the reply, then the "add/show 1 more comment" link appears immediately after I post my comment. However, when I click it, it disappears and nothing is shown.  I don't think it's that OP is posting two comments of which he deletes the second one, because this has happened to me several times, even when I was replying within 2-3 seconds.  I do, however, believe that this is more or less the same bug as , except that I don't have to reload the page in order the link to be gone, and I also seem to be getting the erroneous behavior only if I reply to a comment which is itself a reply to an earlier comment of mine.  Unfortunately, I wasn't careful enough to take a screenshot last time this happened, but here's what I am talking about for those who like visual demonstration better: At the start of A New Hope, when the corvette is being attacked, C3PO says something to the effect of:     There'll be no escape for the princess this time.   But when Luke stumbles onto Leia's message in R2D2, C3PO seems to not know who the figure in the message is:     I'm afraid I'm not quite sure, sir. I think she was a passenger on our last voyage. A person of some importance, sir - I believe.   Is C3PO feigning ignorance to protect her, or does he really not recognize her? Is it appropriate to put a comma before "which"? On Server Fault, we were discussing  in chat. There are non-native speakers in the chat and somehow we got on the topic of grammar and proper use of commas.   The sentence of interest:     I have just got a new Ubuntu LAMP VPS server, which I am trying to configure.   Some people thought that the comma was superfluous. Others did not.   When is it appropriate for a comma to preceed the word which, and is the example sentence above correct?    Also, before you vote to close this question as a dupe of , please consider the differences between the two questions and ignore the similarity of the titles. The OP's example sentence was restructured in that question instead of addressing the proper use of commas. Elementary central binomial coefficient estimates How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}&lt;\binom{2n}{n}&lt;\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ &gt; 1 ?     Does anyone know any better elementary estimates?   Attempt.  We have $$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}=\prod_{k=0}^{n-1}\left(1+\frac{k}{2(n-k)}\right).$$ Then we have $$\left(1+\frac{k}{2(n-k)}\right)&gt;\sqrt{1+\frac{k}{n-k}}=\frac{\sqrt{n}}{\sqrt{n-k}}.$$ So maybe, for the lower bound, we have $$\frac{n^{\frac{n}{2}}}{\sqrt{n!}}=\prod_{k=0}^{n-1}\frac{\sqrt{n}}{\sqrt{n-k}}&gt;\frac{2^n}{\sqrt{4n}}.$$ By Stirling, $n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$, so the lhs becomes $$\frac{e^{\frac{n}{2}}}{(2\pi n)^{\frac14}},$$ but this isn't $&gt;\frac{2^n}{\sqrt{4n}}$. Reputation in summary tab shows as blank instead of 0, if net result is zero Why is negative times negative = positive? Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.  I went ahead and gave them a proof by contradiction like so:  Assume $(-x) \cdot (-y) = -xy$  Then divide both sides by $(-x)$ and you get $(-y) = y$  Since we have a contradiction, then our first assumption must be incorrect.  I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote.  Is there a better way to explain this?  Is my proof incorrect?  Also, what would be an intuitive way to explain the negation concept, if there is one? Determining statistical probability related to a string event of dates Words for "a bit sad" and "a bit happy"? I want to write a query like this:  SELECT o.OrderId, MAX(o.NegotiatedPrice, o.SuggestedPrice) FROM Order o   But this isn't how the MAX function works, right?  It is an aggregate function so it expects a single parameter and then returns the MAX of all rows.     Does anyone know how to do it my way? Movie with humans battling aliens piloting giant animalistic, water-walking robots or sea warships I have memories of watching it a few years ago (I was a kid then) in the early 2000s probably; before 2010 I think. It was a movie with giant robots, and the enemies were aliens piloting what I remember as giant, more animalistic robots. I remember those animalistic ones running on water. The aliens themselves were a little taller than humans but mostly humanoid. I remember what the aliens look like a bit; most of the movie they were clad in some kind of armor, but the scene I mentioned showed one's face, if I remember well, grey skin, vividly green eyes, a beard made of some kind of quills, bald. I have this one scene that stayed crystal clear in my memories. The heroes had captured an injured alien, and other aliens attack to, in the end, just retrieve the injured one, leaving with this one without attacking further this time. I also remember the aliens going through a city in some machine (I'm actually not sure this one was piloted though) and destroying anything it perceived as a threat, from cars to security cameras. My memories are coming from years ago, so I might be wrong in a bunch of things there, but this one scene, I can assure is correct. This might remind a bit of Pacific Rim, but it isn't. (It's watching Pacific Rim that reminded me of this movie.) It doesn't seem to show up on any giant robot movie list on internet, weirdly, so I hope you can help me. And now I'm starting to wonder if there was really robots on the heroes' side. Maybe it was warships (sea ones, not spaceships). As I said it was some time ago and I was young, so bad memories are likely.