How to compute     $$S=\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$$   I tried to rewrite it in terms of $\sin$  $$ \csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}= \frac{1}{\sin^2\frac{\pi}{14}}+\frac{1}{\sin^2\frac{3\pi}{14}}+\frac{1}{\sin^2\frac{5\pi}{14}}=\\ \\  \frac{\sin^2\frac{3\pi}{14}\sin^2\frac{5\pi}{14}+\sin^2\frac{\pi}{14}\sin^2\frac{5\pi}{14}+\sin^2\frac{\pi}{14}\sin^2\frac{3\pi}{14}}{\sin^2\frac{\pi}{14}\sin^2\frac{3\pi}{14}\sin^2\frac{5\pi}{14}}$$  then i used  $$2\cos x\cos y=\cos(x-y)+\cos(x+y)\\ 2\sin x\sin y=\cos(x-y)-\cos(x+y)$$  then i found \begin{align} S &amp;= 2\frac{\left(\cos\frac{\pi}{7}-\cos\frac{4\pi}{7}\right)^2+\left(\cos\frac{2\pi}{7}-\cos\frac{3\pi}{7}\right)^2+\left(\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}\right)^2}{\left(\cos\frac{\pi}{7}-\cos\frac{4\pi}{7}\right)\left(\cos\frac{2\pi}{7}-\cos\frac{3\pi}{7}\right)\left(\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}\right)} \end{align}  i made a simplification and used again the transformation of product on sum and arrived at \begin{align} S &amp;= 4\frac{6-5\cos\frac{\pi}{7}+2\cos\frac{2\pi}{7}-4\cos\frac{3\pi}{7}+2\cos\frac{4\pi}{7}-4\cos\frac{5\pi}{7}+\cos\frac{6\pi}{7}}{5-8\cos\frac{\pi}{7}+6\cos\frac{2\pi}{7}-5\cos\frac{3\pi}{7}+4\cos\frac{4\pi}{7}-3\cos\frac{5\pi}{7}+\cos\frac{6\pi}{7}} \\  &amp; \cos\frac{6\pi}{7}=-\cos\frac{\pi}{7} \hspace{5mm}  \cos\frac{5\pi}{7}=-\cos\frac{2\pi}{7} \hspace{5mm} \cos\frac{4\pi}{7}=-\cos\frac{3\pi}{7} \\ S &amp;= 4\frac{6+6\left(-\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}-\cos\frac{3\pi}{7}\right)}{5+9\left(-\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}-\cos\frac{3\pi}{7}\right)} \end{align}     $$-\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}-\cos\frac{3\pi}{7}=-\frac{1}{2}$$   $$S=4\frac{6-3}{5-\frac{9}{2}}=4\frac{3}{\frac{1}{2}}=4\cdot3\cdot2=24$$
How do I show that:  $$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$  This is actually problem B $4371$ given at this . Looks like a very interesting problem.   My attempts: Well, I have been thinking about this for the whole day, and I have got some insights. I don't believe my insights will lead me to a $\text{complete}$ solution.   First, I wrote $\sin\frac{5\pi}{14}$ as $\sin\frac{9 \pi}{14}$ so that if I put $A = \frac{\pi}{14}$ so that the given equation becomes, $$\frac{1}{\sin^{2}{A}} + \frac{1}{\sin^{2}{3A}} + \frac{1}{\sin^{2}{9A}} =24$$ Then I tried working with this by taking $\text{lcm}$ and multiplying and doing something, which appeared futile. Next, I actually didn't work it out, but I think we have to look for a equation which has roots as $\sin$ and then use $\text{sum of roots}$ formulas to get $24$. I think I haven't explained this clearly.         $\text{Thirdly, is there a trick proving such type of identities using Gauss sums ?}$ One post related to this is:  I don't know how this will help as I haven't studied anything yet regarding Gauss sums.
How to debug "Cross-thread operation not valid" error in C#? How to "find and replace" in a PDF on OS X? ReaderWriterLockSlim vs. Monitor I have an IDictionary&lt;TKey,TValue&gt; implementation that internally holds n other Dictionary&lt;TKey, TValue&gt; and distributes that insertions by the HashCode of the key to the invidual sub-dictionaries. With 16 sub-dictionaries, the number of collisions is pretty low on a 4-core machine.  For parallel insertions, i locked the Add-method with a ReaderWriterLockSlim, locking only the individual sub-dictionary:    public void Add(TKey key, TValue value)         {             int poolIndex = GetPoolIndex(key);             this.locks[poolIndex].EnterWriteLock();             try             {                 this.pools[poolIndex].Add(key, value);             }             finally             {                 this.locks[poolIndex].ExitWriteLock();             }         }   When inserting items with four threads, i only got about 32% cpu usage and bad performance. So i replaced the ReaderWriterLockSlim by a Monitor (i.e., the lock keyword). CPU usage was now at nearly 100% and the performance was more than doubled.   My question is: Why did the CPU usage increase? The number of collisions should not have changed. What makes ReaderWriterLock.EnterWriteLock wait so many times? Suppose that G is a group and N &lt; G a normal subgroup. Assume that there is no normal subgroup M &lt; G with N &lt; M. Prove that G/N is simple.  This problem is really throwing me for a loop. The concept of quotient groups and how they relate to simple subgroups is confusing me, and so I am not really sure how to begin. If a simple subgroup's only normal subgroup is itself (and the trivial group), then G/N being simple would be proving that when multiplying the entire subgroup N by an element g for all g in G, that the collection of those sets cannot be broken into smaller normal subgroups? I changed both the desktop and login screen background photos. However,only desktop photo is changed. The one on the login screen is still the purple Ubuntu one. Is this a bug? I would like a way to convert a document from HTML to LaTeX on a Windows platform. What is my best option? bash: ./program: cannot execute binary file: Exec format error I'm trying to run a program, but an error happens like this:  bash: ./program: cannot execute binary file: Exec format error   The result of file program was:  program: ELF-32-bit LSB executable, ARM, EABI4 version 1 (SYSV), dynamically linked(uses share libs), for GNU/LINUX 2.6.16, not stripped   How can I fix this error?  I'm using Ubuntu 14.04.2 (amd64) with VMware. I also tried with Ubuntu i386, but result was same. What's with the Japanese SO's stats? I want to run a command when there are only 10 entities left. I know you can use /execute unless entity to detect when all matching mobs have died, but it doesn't work if you want to run a command when there are for example 10 entities left.  A command like /execute if entity @e[limit=10] does not work, it also triggers if there is only one entity.  Such a counting system could also be used to determine once there is only one player left in an arena, meaning they have won a battle or similar. If $f'$ is bounded, then $f$ is uniformly continuous Let $f\colon(a,b) \to \mathbb R$ be differentiable and suppose that there exists an $M&gt;0$ such that $|f&#39;(x)| \leq M$ for all $x$ in $(a,b)$. Prove that $f$  is uniformly continuous. Filtered Review Queue doesn't show any audits How to detect the Internet connection is offline in JavaScript? A proof using Yoneda lemma New Zealand pronunciation of "women" vs "woman" I have read in a number of places that the NZ pronunciation of "women" must be rather peculiar. Quoting from :     For some years I've noted the tendency of Kiwis to pronounce "woman" and "women" identically (as "woman").   To which an Australian replies:     It appears there is a vowel shift going on in New Zealand (NZ) English [...] [T]he vowel in "women" which in Australian English is the same as the vowel in "hit", is often, in NZ, reduced to a schwa [...]. It makes the plural sound like our singular. I don't think the NZ pronunciation of the singular is the same as the NZ plural   A New Zealander begs to differ:     I've noticed this trend over the last few years also. To me as a kiwi it doesn't sound anything like a mispronounced plural -- it just sounds as if the speaker is using the one word for both singular and plural. [...] I see it as part of a much larger tendency to confuse singular and plurals. [Several examples follow.]   Who's right? Who's wrong? Wikipedia has :     In New Zealand English the short i of KIT is a central vowel not phonologically distinct from schwa /ə/, the vowel in unstressed "the". It thus contrasts sharply with the [i] vowel heard in Australia. Recent acoustic studies featuring both Australian and New Zealand voices show that the accents were more similar before the Second World War and that the KIT vowel has undergone rapid centralisation in New Zealand English. Because of this difference in pronunciation, some New Zealanders claim that Australians say "feesh and cheeps" for fish and chips while some Australians counter that New Zealanders say "fush and chups".   So, there appears to be a vowel shift going on; but I'm not sure if there might be some truth to the people-just-stop-caring-about-certain-plurals argument as well.  I hope we have enough New Zealanders on board to shed some light on this.   Can you provide a guide to how to pronounce "women" and "woman" in NZ English? (I've done some searching for audio files or YouTube videos, but haven't found anything yet.)  Would you say that it's just the vowel shift, "a much larger tendency to confuse singular and plurals", or both?  How recent is this trend? (E.g., if you are a native speaker of NZ English, is your pronunciation of "women" different from that of your parents/grandparents?) How can I reliably get the operating system's name? Say I am logged into a remote system, how can I know what it's running? On most modern Linuxes (Linuces?), you have the lsb_release command:  $ lsb_release -ic     Distributor ID: LinuxMint Codename:       debian   Which as far as I can tell just gives the same info as /etc/lsb-release. What if that file is not present? I seem to recall that the lsb_release command is relatively new so what if I have to get the OS of an older system?  In any case, lsb stands for Linux Standard Base so I am assuming it won't work on non-Linux Unices. As far as I know, there is no way of getting this information from uname so how can I get this on systems that do not use lsb_release? Suppose there are $2$ permutations given by $p$ and $q$. 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(Note: Just looping over all values of $i$ is not efficient as $i$ may be very large) Edit: Another way I can think of solving this is by writing both $p$ and $q$ as product of disjoint cycles: $$ p = c_1c_2\ldots c_k \text{ and } q = d_1d_2\ldots d_l $$ And thus, as disjoint cycles commute: $$ p^i = c_1^ic_2^i\ldots c_k^i $$ How to proceed after this? Projecting sp objects in R I have a number of shapefiles in different CRSs (mostly WGS84 lat/lon) that I'd like to transform into a common projection (likely Albers Equal Area Conic, but I may ask for help on choosing in another question once my problem gets better-defined).  I spent a few months doing spatial stats stuff in R, but it was 5 years ago.  For the life of me, I cannot remember how to transform an sp object (e.g. SpatialPolygonsDataFrame) from one projection to another.  Example code:  P4S.latlon &lt;- CRS("+proj=longlat +datum=WGS84") hrr.shp &lt;- readShapePoly("HRR_Bdry"), verbose=TRUE, proj4string=P4S.latlon)  # Shapefile available at  #   http://www.dartmouthatlas.org/downloads/geography/hrr_bdry.zip  #   but you must rename all the filenames to have the same  #   capitalization for it to work in R   Now I have a SpatialPolygonsDataFrame with appropriate projection information, but I'd like to transform it to the desired projection.  I recall there being a somewhat unintuitively-named function for this, but I can't remember what it is.  Note that I do not want just to change the CRS but to change the coordinates to match ("reproject", "transform", etc.).  Edit  Excluding AK/HI which are annoyingly placed in Mexico for this shapefile:  library(taRifx.geo) hrr.shp &lt;-    subset(hrr.shp, !(grepl( "AK-" , hrr.shp@data$HRRCITY ) |                                      grepl( "HI-" , hrr.shp@data$HRRCITY )) ) proj4string(hrr.shp) &lt;- P4S.latlon How to read value of variable, where the name of variable is the value of another variable I watched a video on youtube (I lost it) that shows rendering parts of the screen to combine them in a single image. I followed it to the point where I couldn't adjust the dimensions under the "set render border", but in the video he had dimensions and I have nothing.  I need it for 4K and 8K renders (without anti-aliasing), but my laptop can't handle it all at once and I don't have time for a 4 hour waiting so this feature will work perfectly.  I'm using version 2.76 so is it moved to another place or removed. Maybe a bug? 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