My teacher told us that : $$\sum_{k=0}^{n-1} \cos\frac{2k\pi}{n}=0$$ Without giving any proof If someone could tell me how to approach this that would be great !
Possible Duplicate:          Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\alpha + (n-1)\beta] = \frac{\cos(\alpha + \frac{n-1}{2}\beta) \cdot \sin\frac{n\beta}{2}}{\sin\frac{\beta}{2}} $$
Highlight tags corresponding to our favorites Each of us have our list of favorite tags.  When a question has tag(s) that match one or more of ours, the background gets highlighted, drawing our attention to the question.  It would also be nice to highlight (perhaps using a bold font) the tags(s) that matched our own, so one could quickly do a further filtering of the question to see if it is of interest at the present time, like the following:    e.g. arduino is in my list of favorites, but the other two in each question are not. Label equations arbitrarily When I have equations, as in  \begin{equation} ... \end{equation}   Can I arbitrarily assign the numbers to the equations? Instead of the usual order (1), (2), (3), ..., I'd like something like (1), (2.1), (2.2), (2.3), (3.1), (3.2), (4), ... convert bash `ls` output to json array Where do I put the code snippets I found here or somewhere else on the web? Many posts here or somewhere else contain code, but they do not say where to put it.  Example:  I have found this post:  I'm a newbie with PHP. Where exactly should I place the code from the answer? Can I use the Adwaita theme in Unity without installing GNOME shell? I recently installed Ubuntu 11.10 (new install). I would like to use the Adwaita theme with Unity, but it seems that it is not available in the theme selection dialog. I know I get it there after installing gnome-shell, but is there a way to use Adwaita theme for Unity without installing gnome-shell? Footnotes in tables? I would like to have footnotes in tables, but the \footnote command does not work inside a table environment.  Googling one finds that this is , at least in the UK.  That link contains a list of solutions to the problem, but since the author of the FAQ does not believe that this is good style, no comparison is made between the different methods.  I am sure that there are good typographical reasons why in most situations it is undesirable to have footnotes in tables (although I have never come across them), however I have recently felt the need to do just this in a paper I am currently writing.  Therefore I'd like to ask two questions:   What are the arguments against footnotes in tables? If one really must go against the advice in 1., which method (whether considered in the  or not) would you recommend?   Thanks in advance.  Added  In response to ShreevatsaR's comment: I do not much care for where the text of the footnote appears.  I do however insist on the footnote marks being attached to the relevant entries in the table, so that I can avoid having to reference "the second entry in the eighth line of the table", etc... How to construct a relative path in Java from two absolute paths (or URLs)? Given two absolute paths, e.g.  /var/data/stuff/xyz.dat /var/data   How can one create a relative path that uses the second path as its base? In the example above, the result should be: ./stuff/xyz.dat I am trying to prove the following statement but have trouble  comprehending/going forward with some parts! Here is the statement:     If $A$ is any set, then  $|A|$ $&lt;$ $|P(A)|$   Here is what I have so far:  We need to show that there is an injection from $A$ to $P(A)$ but not a surjection.   A natural choice for an injection is the function $ f(x)$ $=$ $\{x \}$, which in plain English, takes any element $x$ (that is in $A$) and sends it to the one-element set $\{x \}$. Thus $f(x)$ is injective!  To show that there is no surjection, for the sake of contradiction, assume there is a surjection. Here is where I start to have trouble. Surjectivity means that every element of the co-domain is mapped to an element of the domain, correct? Consequently, in this particular case, we are "matching" sets (from $P(A)$) to elements (from $A$) right?  If the above is correct, my problem arises here. I am not sure how to prove that $f$ is not surjective. Unfortunately, I am easily confused by notation so please explain in English.  Thank you in advance!! :) "To be subject to" vs. "to be subjected to" I read an article from  today which stated:     TTC workers are subject to alcohol and drug testing.   A later paragraph of the same article repeated it, except it used subjected to instead of subject to.  I have rarely seen the use of subjected to before. Is it legitimate in English as well? Why didn’t the author just keep it consistent by using subject to throughout the article? Does using varieties of forms enrich the expressions within an article? If verification is quality control, what is a validation? First, a word about the format - I don't know whether this was a movie or a TV series, b/c I saw it aired in pieces on PBS back in the 1980s. I don't even recall if it was originally voiced in english, dubbed over, or subtitled. I'll just call it a movie for now.  The movie seemed to be set in Europe in what looked like the 1960's or 70's, where almost all the people were gone. The protagonist was probably a 20-something man, who routinely visited a psychiatrist. I think the protagonist also had a girlfriend, but most of the dialogs seemed to be between the man and the doctor.   It was kind of a mystery where all the people had gone, and I think the man was nominally seeing the doctor to cope with the stress of it all, or had amnesia and suspected he knew The Truth. However, it turns out that the doctor discovers that the man is somehow the key to the reality that the characters share, and during hypnosis sessions, discovers he can manipulate the man and thereby reality. To make it easier for himself, the doctor gives the man a code word to return the man to a hypnotic trance on command. The code word, I am positive, was 'Antwerp'.  I don't think I ever saw the end of this, but the 'truth' might have been that World War III was fought with neutron bombs, and that the man was actually just dreaming everything as he wandered around the city, dying. Or not. It would be nice to know how it turned out, but I don't even have the name of the show. Fibonacci proof question: $f_{n+1}f_{n-1}-f_n^2=(-1)^n$ Why is Tony Stark in the Iron Man suit unaffected by G-forces, inertia and blunt force trauma? Tony's suit is an engineering marvel, and they explain many facets of the technology to help give legitimacy to the special powers, abilities and weapons it has.  I never read the Iron Man comics, but in the movie they really don't have a technological explanation for how he miraculously seems to be unaffected by crushing G-forces of sudden acceleration, the inertia of sudden stops and hair-pin turns, and especially the blunt force trauma from getting smashed in the face with Thor's hammer.  Certainly it appears the suit takes punishment, but Tony always remains unscatched, never a collapsed lung, broken bone or even a bloody nose.  In my infantile understanding of theoretical physics, the same technology that would be used to counter G-forces and sudden blunt force trauma to the body inside would be the very same technology used for  which in my honest opinion would be a much more stunning technological achievement than his heralded Arc Reactor.  Was their ever an explanation proferred for this and if so was it indeed attributed to Anti-Gravity? From injective map to continuous map How do I tell Finder to always show my Library folder on Mavericks? I thought there was a checkbox to suppress the hiding of my user Library folder on Mavericks, but I can't seem to find it. I appreciate hiding Library by default, but  have missed it being easily accessible.  How can I get the library folder to be visible in Finder, the Finder go menus / file save dialogs without needing to hold down shift? `find` with multiple `-name` and `-exec` executes only the last matches of `-name` Is the uncertainty principle a property of elementary particles or a result of our measurement tools? In many physics divulgation books I've read, this seems to be a commonly accepted point of view (I'm making this quote up, as I don't remember the exact words, but this should give you an idea):      is not a result of our lack of proper measurement tools. The fact that we can't precisely know both the position and momentum of an elementary particle is, indeed, a property of the particle itself. It is an intrinsic property of the Universe we live in.   Then this video came out:  (skip to , if you're already familiar with the uncertainty principle).  So, correct me if I'm wrong, what we may claim according to the video is:     the only way to measure an elementary particle is to make it interact with another elementary particle: it is therefore incorrect to say that an elementary particle doesn't have a well defined momentum/position before we make our measurement. We cannot access this data (momentum/position) without changing it, therefore it is correct to say that our ignorance about this data is not an intrinsic property of the Universe (but, rather, an important limit of how we can measure it).   Please tell me how can both of the highlighted paragraphs be true or how they should be corrected. Shapefile: "table1.shp"   Field name: "Name" Field name: "Result"   Flat file: "referenceTable"   Field name: "STREET" Field name: "AKA"   I need to compare the value "Name" in "table1.shp" to the value "STREET" in "referenceTable"  If the values are the same then put the "AKA" value into the "Result" in "table1.shp"     Here is my code.  I got stuck on accessing the elements from the array  import arcpy, os, sys, string from array import *   table1 = "G:\\xStreetNew\\table1.shp" table1_column = ['Name', 'Result'] referenceTable = "G:\\xStreetNew\\Test.gdb\\referenceTable" referenceTable_column = ['STREET', 'AKA']  referenceTable_List = [] with arcpy.da.SearchCursor(referenceTable, referenceTable_column) as rCur: for r_row in rCur:     reference_row = r_row[0].split(",")     referenceTable_List.insert(0,reference_row) ##print sorted(referenceTable_List)  Table1_Update_Cursor = arcpy.da.UpdateCursor(table1, "Name") for row in Table1_Update_Cursor:   if row[0] == None:   row[0] = row[0] else:   row[0] = row[0].replace(";",",")   Table1_Update_Cursor.updateRow(row)  table1_List = [] with arcpy.da.SearchCursor(table1, table1_column) as tCur: for t_row in tCur:     row = t_row[0].split(",")     table1_List.insert(0,row) ##print sorted(table1_List)  Table1_row = len(table1_List) Table1_column = len([table1_List]) ref_row = len(referenceTable_List) for i in range(0,Table1_row):  #Table Row   for ii in range(0,Table1_column):   #Table Column     print table1_List[i][ii]   Below was my old post.  Sorry for the confusion.  I have a shapefile named "table1.shp".     I would like to get all values from fielname "Name" from this table and put them into a list. Able to output each element from the list     So far, I'm able to create a list but not too sure how to get to value "AVENUE 33" or "CHAPMAN ST"  This is my code:  import arcpy, os, sys, string from array import *   table1 = "G:\\xStreetNew\\table1.shp" referenceTable = "G:\\xStreetNew\\Test.gdb\\referenceTable" cursor = arcpy.da.UpdateCursor(table1, "Name") for row in cursor: if row[0] == None:     row[0] = row[0] else:     row[0] = row[0].replace(";",",") cursor.updateRow(row) table1_column = ['Name'] table1_curObj = arcpy.da.SearchCursor(table1, table1_column) table1_List = [] for row in table1_curObj:    table1_row = row[0].split(",")    table1_List.insert(0,table1_row)    print table1_List[0] Pre-image of a measurable set A is always measurable? Partition a set into subsets of size $k$