how to prove $\left( {\frac{1}{\sin z}} \right)^{2}=\sum_{-\infty}^{+\infty}\frac{1}{\left( {z-\pi n} \right)^2}$
Prove that $\frac{1}{\sin^2 z } = \sum\limits_{n= -\infty} ^ {+\infty} \frac{1}{(z-\pi n)^2} $
My Windows folder takes up too much space - how can I reduce it? In typical PCs the Size of Windows Folder is 6 to 8 GB but in my Case its 25GB. How can I delete the unwanted files from the Windows Folder (which, in turn, will optimize my C Drive size).  My system is Windows 7 with 60 GB in C Drive with 59 GB occupied. Is there any online resource that can tell me what visa rules are for passport holders of a specific country? When thinking about travelling, I often have to think about a place to visit, and then look up visa rules for that particular country. This can be a fairly painful process because official government sites can often be hard to find, and advice on (say) guidebooks or Wikitravel tends to be generic.  Is there any resource where I can enter my nationality/residency, and then look up a list of visa rules for various countries? Correct way to render crepuscular rays in cycles? While trying to use the new cycles volumetric rending features to create  (with the intention of updating ), I have found this to be a little trickier than I anticipated.    When I create a basic setup with a spot lamp (sun lamps don't seem to work, not sure if that is a result of their one directional nature or what..) and a world material like this:    Which gives me this (500 samples):    The problems with this that make me think that I'm missing something (aside from the rendertime and noise, which I understand to be a part of using bleeding edge new features) are:   The background shader/sky texture isn't visible at all (volume is too dense?) The lamp needed to be set to 50000 strength (this is of course dependent on the distance from the camera, but the kinds of strengths required to get some nice defined rays will blow everything out if volumes are disabled)   I can't seem to get a straight render (no compositing tricks) to look anything remotely like the pretty image in the release notes:        (source: )     Is this just how it works? Or is there some trick I have missed (not counting compositing/renderlayer tricks)? Disabling Intel Turbo Boost in ubuntu Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$ I need your help with evaluating this limit:  $$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n,$$  i.e. we apply the $\sin$ function $n$ times.  Thank you. Companion affinity change notification Common lines between two files , where a user had taken the time to look at the link on a post and realized that the post wasn't just off-topic, it was actually half-decently-disguised spam. So he flagged it as spam, and an overworked mod declined it, giving the questioner the benefit of the doubt. And we probably want mods to be giving the questioner the benefit of the doubt.  We also want to get rid of spam ASAP, and make the lives of mods easier. I'd like to suggest a feature the mod in question mused might be useful: An optional textbox where you can add a note saying why you think it's spam, so the mod can see it easily. In this case, the helpful user who went the extra mile could have written "Site linked doesn't relate at all to the question, instead solicits BitCoins" or similar.  One can leave a comment on the question, but:   People sometimes leave...unhelpful...comments on spam questions, and we don't want mods to have to take the time to read through them hoping for something useful It allows the user flagging the spam to do so and say why without calling attention to themselves publicly     Update: We may well want this for all flags, but in particular I had reason to want it for an NAA flag, so I've posted . Roots of $\,ax^2+bx+c\,$ are irrational if $a,b,c$ are odd integers How would I show the following.  For all odd integers a,b, and c if z is a solution of $ax^2+bx+c=0$ the z is not rational.  How would I show this? This is what I did  Let a be an odd integer then for a integer k it has the form $a=2k+1$  then $b=2t+1$ for if c integer.  then $c=2r+1$ if s is an integer then I do  $(2k+1)(x)^2+2t+1(x)+(2r+1)=0$  But I find my self stuck on what to do any hint are appreciated. I tried to find this question somewhere else on here, and found a couple that were similar to it -- but all the answers seemed pretty convoluted, and I think it was surprising to me because I just imagined that SQL would have a way easier way to do this.  Essentially, I'm using a GROUP BY to group two values on top of each other. The coding is simple and works well, where I get these results using WITH ROLLUP at the end:  Type    Person  Count Type A  Person 1    10 Type A  Person 2    91 Type A  Person 3    13 Type A  Person 4    10 Type A  Person 5    2 Type A  NULL        126 Type B  Person 6    16 Type B  Person 7    63 Type B  Person 8    15 Type B  Person 9    22 Type B  Person 10   16 Type B  Person 11   3 Type B  Person 12   20 Type B  Person 13   44 Type B  NULL        198 NULL    NULL        360   Is there an easy way to substitute something saying if "NULL" is in the "Person" column, then make NULL AS "Subtotal" and if "NULL" is in the "Type" column, NULL AS "Grand Total?"  And if that's weirdly complicated, I'd be totally fine with just calling all of them "Total."  Thanks! Variance of product of dependent variables When handling multiple white-on-transparent png images, I am hindered by the fact that the thumbnails of these images are all-white. Both Adobe Bridge CS6 and Windows Explorer suffer the same problem, I can't discern between two of these pngs.  Is there a way to make these thumbnails useful? Is there some setting I missed that allows changing the thumbnail background colour for a png -- now a deafult white? Why not to extend the set of natural numbers to make it closed under division by zero? We add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers).  Why this trick isn't performed with division by zero? If the Universe is expanding, how is that galaxies run into each other? As the title says. If everything is moving outward, thrown by the Big Bang, how could two Galaxies end up with velocity vectors that run them into each other? I have Ubuntu 16.04 LTS installed on my PC. I have a problem playing  videos on google chrome. Version of google chrome: 51.0.2704.63 (64-bit). All other videos of youtube are playing normally in google chrome. I even tried playing the  videos on Firefox but the same problem persists. Can anyone help me with the same ? Booting a Windows 7 installation on different hardware I'm in a situation where I could do with very quickly migrating a Windows 7 (RTM x64) installation from one machine to another.  What options are open to me in terms of getting W7 to boot after the drive is picked up and moved from one box to another? I thought it was supposed to be a little less sensitive to this kind of move than XP, but it doesn't work - it is stuck in a reboot loop and never reaches a GUI.  Tried a few things so far, none of which have worked:   Changed SATA mode in the BIOS of the target machine between ATA and AHCI run Windows 7 Startup Repair tried safe mode, no change   (I will keep this list up to date as suggestions come in)  UPDATE: I can confirm this reboot loop is due to STOP error 0x0000007B, and these codes follow:  0xFFFFF880009A98E8 0xFFFFFFFFC0000034 0x0000000000000000 0x0000000000000000    UPDATE: I didn't get anywhere on this and I ended up just rebuilding the machine. I think it should be theoretically possible, so I'm going to leave the question open in case someone comes along in future with an answer. Training N classifiers for N labels vs one classifier with N labels I have a classification problem which is multi-label with N labels. I would like to know which method would be the better choice? Training N classifiers (1 for each label) or a single classifier which use K-hot encoding. I'm using a neural network for this with a softmax at the last layer. (Is there any chance that the softmax would cause a problem?) I'm using tensorflow for the model. The LaTeX Wikibook :     Please understand: you do not have to use floating tables. It only   looks more professional. If you want to place your tables where they   lie in your source code, do not use table at all! This is a very   common misunderstanding among newcomers.   LaTeX works with an intricate , but I often prefer immediate placement with the H placement specifier (enabled by the float package).  My understanding is that captions and table/figure numbering require the table/figure environments. Is this really so, or are there ways around that? What other reasons are there for using table/figure if we know that we don't want a particular object to float? Does Sheldon Cooper have Asperger's Syndrome? Replace $ with £ using cat Command I have file test_output_archive.svd with some Dollar signs ($). I have generated a shell script Replace_dollar_pound.sh with cat command to replace dollar symbol with Pound (£) script is working on Solaris 10, Dollar getting replaced as expected. In Solaris 11, but the dollar is not getting replaced, not throwing any errors  Content of test_output_archive.svd : Line Rental $9  OOB Only for $10  Replace Dollar symbol with Pound at step1, Remove junk charactes at step 2, Remove temp file at step3, Rename at step4**  Content of Replace_dollar_pound.sh : cat test_output_archive.svd | tr "\044" "\243" > temp_archive.svd cat temp_archive.svd | tr -d "\302" > test_output_archive2.svd rm temp_archive.svd mv test_output_archive2.svd test_output_archive.svd  Any Modifications to be done as OS is updated to Solaris 11 ??