Prove that if n is a positive integer, then $\left|\frac{sin(\frac{n\theta}{2})}{sin(\frac{\theta}{2})}\right|\le n$ for $(θ ≠ 0, ±2π, ±4π, . . .)$ Any hint on how to start this question will be appreciated.
I was trying to prove following inequality:  $$|\sin n\theta| \leq n\sin \theta \ \text{for all n=1,2,3... and } \  0&lt;\theta&lt;π $$  I succeeded in proving this via induction but I didn't get "feel" over the proof. Are there other proof for this inequality?
Found this when subscribing to jQuery tag. "unsubscribe | rss" is not shown correctly. Say my opponent plays a planeswalker during their first main phase.  I have a  or some other effect that will kill it after it is already in play, but before they use any of its abilities.  I could respond to them using a +1 ability, but I want to kill it before that effect goes on the stack.  In most casual situations, it would basically come down to whether I say "Hero's Downfall" first, or they say "+1" first.  But I can't imagine that's how it's supposed to work.  Is there a time between when the planeswalker hits the battlefield and when they activate its abilities that they must let me respond? It seems to me that situations like this arise frequently, and they are resolved simply by who speaks up first, which is horrible for such a rules-oriented game. Using Either a Comma or a Semicolon My teacher was explaining how sometimes a comma isn't enough to make a sentence correct, and that sometimes you have to use a semicolon. I am having trouble figuring out when you should use which.   Can someone please explain? I was trying to install Ubuntu 12.04 in VirtualBox 4.2.12r84980. I see    this kernel requires an x86-64 CPU, but only detects an i686 CPU, unable to boot   But I am using a 64 bit Windows 8, and trying same .iso for trying Ubuntu. Then what is the problem? Density of $\mathbb{Q}^n$ in $\mathbb{R}^n$ Good color palette, usable for the color blind. I'm designing a chart right now, that will slowly have more items added to it. I'd like to create an list/order of colors for the items, that will be most usable for the color blind to differentiate between each item. Any ideas how/where to start? Adjective for "terrified yet calm and capable" What is an adjective to describe the duality of a first responder (such as an EMT responding to a terrorist attack) who is terrified for the wounded, yet simultaneously calm, detached, and capable enough to give aid? So, I have about 7 actions users may need in my app, and I have to use text instead of icons because it is a customer requirement.  I can add only 3 actions to my toolbar on iPhone 6. On Android you have the hamburger menu (see image), but I have never seen it in iOS.       So, what is the appropriate concept in iOS to add many actions into a menu? If friction acts on car all time, why is it not accelerating forever? In the first gear, slowly releasing my clutch i push the gas pedal to the maximum torque level, say at 2000 rpm. Now as the engine torque is acting on the wheel, friction helps the car to accelerate.   If i hold the gas pedal at that position, the torque should continuously act on the wheel and the car should increase speed forever. But in reality even if we put the gas pedal at the max.torque rpm range in first gear it speeds up to certain wheel rpm (say 500rpm) and it stays there only. Why? Over 20GB of disk space is gone? System drive, OS is Windows 7 x64 SP1 Ultimate.    Windows Explorer shows 100GB of used space.  TreeSizeFree shows 75GB of used space.  Total Commander and FAR confirm TreeSizeFree's measurements (counting hidden files).  Shadow copy is using up to 5GB with the current usage of 1.36GB.  Running FAR from Win7 PE shows the same.  Checkdisk shows no bad sectors. Disk is an OCZ SSD.  SMART monitor suggests SSD condition is perfect.    Where is the rest of space gone? Hey I want to connect the bottom loop with the top loop so that it forms a solid.  I have tried selecting multiple vertices and pressing F but that just creates a large face which I do not want. I have tried using grid fill bu that doesn't work either. The solution that seems to work is by selecting four vertices and pressing F and then selecting the next 4 and pressing F and on and on. That just takes too long. How to fix a grub boot error : "symbol 'grub_calloc' not found Just ran the latest batch of updates on 20.04 (Xubuntu), and now I'm getting a GRUB error: symbol 'grub_calloc' not found  I'm dropped into the 'grub rescue' shell, but have no idea what to do there that might be useful. To me, 'symbol not found' implies some sort of build error with the grub package, but I don't really know how grub works. I noticed that this update also included 'firmware', not sure if that could be related. Is my best bet just to boot from a live CD and see if I can roll back the update to grub somehow? Edited to add: OK, thanks to lots of people ! Here's what I think I now understand.  On 'non-UEFI' systems, grub is installed in two separate parts. The first, most basic, part is the part that is started on bootup. But for most of it's functionality, it needs the second part. These parts must be aligned - neither part must require any functionality from the other part which is not actually there. The visible, run-time problem occurs when these parts are not aligned, and the function grub_calloc is not supplied. It's not 100% clear to me if grub_calloc belongs in the second, larger part or the first. I would have expected the second, but the grub build system is a work of considerable art, so I don't know :).  The root cause of the problem is that the grub update has not ensured that both parts have been updated. Ideally, failure to do this should cause grub installation to fail, and the system should be reverted to a safe state. This does not happen. This is actually still a bit of a mystery to me. All that the update needs to do by default is put each part where the current parts are, because obviously that worked. If the install locations/drives are configuration - driven, and one of these locations can't be reached, then somehow a mismatch has arisen between that configuration data and reality. This might not show up as a problem as long as no new dependency was introduced between the parts.   All flavours of solution involve reinstalling grub to ensure that the two parts are aligned. It's not actually necessary to go back to the previous version ( although that will work ), because it's not the grub runtime per se that is broken. There are numerous ways to achieve this, depending on your environment, but running the Boot-repair live disk worked for me. It may be useful, for the purpose of avoiding such a misalignment in future, to ensure that the grub installer on your system is configured to install to the correct devices. This update resolves some important bugs (). If you have reverted grub to resolve this problem, be aware that you are exposed to these issues. Identify story: Overpopulation problem causes people to live one day a week This story is about a man that is able to live every day of the week, while everybody else has a single day of wakefulness assigned to them. Due to overpopulation, each human spends 6 out of 7 days of the week in stasis, and has jobs assigned to them in daily shifts. 1/7 of the population works on Monday. 1/7 works on Tuesday, &amp; so on. The main character has found away to circumvent the process and illegally live out his days consecutively by skipping the stasis program that is imposed on the general population and by working a different job everyday. Can somebody name this scifi story? How Do Dementors Communicate with Wizards? Run script as root at login (no sudoer user, shell session) Much like /etc/profile and ~/.profile but ran by root instead of the user is doing login. /etc/rc.local runs after boot but I need running the script before login. The user is not a sudoer one.   Thanks! How to convert string to byte array in Python When you reach 10k rep, you get to see other users deleted answers. I just think it's dishonest as a system to call a feature "delete," but show that to totally random strangers in the backroom. Due to the nature of the rep, given enough time, all of the regular users would eventually reach 10k rep, no matter how insensitive or jerk they are.  There's usually a good reason why people hit "delete" to their own work. They regret writing something totally bullshit or maybe they spoke too much about their own work that they shouldn't have.  First of all, is there really a practical reason why 10k rep user should view self-deleted answers? Does the benefit really outweigh the creepiness? (Imagine if total strangers could see your deleted messages in Gmail?)  What suggestions could we make to change it? Borrowing from the email metaphor, I think it makes sense for users to "empty trash" so the items gets seriously deleted. Imho, however the deletion is implemented is not important. It could be a soft delete, SQL Delete, or physical destruction of metal plates as long as the user or the 10k rep user no longer sees the deleted item.  What are the workarounds, if we can't change the behavior?  The least we could do is stop calling it "delete" if it's actually not a "delete." Call it "hide from public" or "actually not delete" or something.  Related user voice:        Edit: Recovery of a deleted-but-good answer is a practical benefit for the community, but I see that as creepy dangerous feature. IMHO, each user should have the rights to take back his or her perfectly good answer, and that the protecting privacy outweighs the benefit or creepiness.   I understand that it's all possible to find things out from Google cache or wherever, but why promote this as default behavior? As 10k rep user, I see others deleted answers by default. @codinghorror wrote:     But like any version control system, deletions are illusory.   Anything and everything that's implemented by human is more or less illusory including phone system and the medical records. Should the government start tapping into phone lines and scanning for words? Remember, this is a question of should had we given the opportunity to design a system. What are the powers shown by the Rings of Power? out file with multiple criteria in python? In , I used quotes to demark specific terms, but after I posted the question, I wondered if I'm doing it right.  Usually, when quoting something a person says, the quotation mark goes after the comma or period, like so:     "This is my opinion," she said.   However, I sometimes use quotes not to represent something said by anyone, but instead for delineating a set term or phrase to make it clear that those words go together and none of them should be mistaken for a part of the rest of the sentence. Like this:     Should I say "stand up", "standup", or "stand-up"?   This looks weird to me:     Should I say "stand up," "standup," or "stand-up"?   ... but maybe I'm wrong about the rules.  Is it correct to use quotes in this way? And if so, are the quotes positioned correctly relative to the other punctuation marks?  Are there other alternatives for delineating terms without the use of quotes?