How to use short stories to explore a new setting and potential characters for a novel?

Question: I'm an aspiring fantasy and sci-fi writer and I my goal is to write a novel that is to be the first in a series within the same setting. I'm getting to a point where I can feel "the world coming together" and images are floating around in my mind. I know what I want my book(s) to be about (as in overarching motives) but don't have characters yet. Would you suggest writing short stories first to get a feel for the setting and explore a couple ideas for characters? I'm asking because I myself have never been interested in reading short stories and I keep telling myself that if J. K. Rowling jumped into her first novel successfully without prior training, I can do it, too. For now I'm limited in the time I can devote to writing so a short story becomes more attractive than plunging into a novel headfirst because finishing it seems that much more achievable.
Comment: To be honest, I've tried doing this by myself, and it didn't help at all. What did help, though, was talking with someone and writing both of out out-of-this-world ideas down.
Comment: Great question, and some brilliant answers. While I can't add anything that hasn't already been said, I would like to provide a couple of examples. Brian McClellan's Powder Mage series has a number of smaller novella spinoffs that are used to explore character or world background (and done so brilliantly), as does Will Wight with his Traveller's Gate series. They can be brilliant tools to flesh out characters or work through interesting story lines that just didn't fit into the main plot (but do serve to enhance the entire mood).
Answer: I'm tempted to tell you that writing some short stories is the "right" way to proceed, but the reality is there is not "right" or "wrong" on this issue.  You need to find what works for you.
The most common path to getting the first full-length novel completed involves plowing right into it with little idea of what you are doing, who your characters are or why you are willing to waste vast spans of time filling pages with drivel.  Then after a week or a month, with rage and self-hatred in abundance, to throw those first pages into a box and ignore them for a time.  Then, when the itch eventually returns,open the box and get back to work.
At that point you have two options.  Throw out what you've already done and now hate, or try to fix it.  In either case, in a month or so, you will have a slightly larger pile of paper which you like a little bit more than the original, but you will have wandered down into a plot hole and gotten stuck about with what to do next.  Back into the box goes all your hard work.
When the itch returns as it always does, you will decide that it is time to seek advice on how to survive this insanity.  Writing books, magazines, groups and forums will become your new obsession.  You will learn about plot design and character creation and how to attack writer's block.  You will start to tool up for the long road ahead.

Maybe you will outline your story down to the smallest detail.
Maybe you will develop each of your characters to a point where they can act on their own with only occasional references back to your plot.
Maybe you will combine those methods into something uniquely your own.

Somehow you will find for yourself a method for putting quality words down on paper and then assembling those pages into art.
Whether you write short stories or epics is not that important.  That you keep writing is.  Some people win the lottery with the first ticket they've ever purchased, but I wouldn't bet my future on either the lottery or the first time win.  Better to learn how to push the pen across the page, even when it doesn't want to go there.  Build your skills and your novel will come.
Find your voice.
Learn your methods.
Feed your muse.
Keep Writing!
In retrospect and realizing that I didn't actually answer your question during that rant, here are the advantages of short stories over longer works...

If writing quality remains constant between the two, short stories are easier and can be completed quicker than long stories.  This allows you to experience the entire literary life cycle in a shorter time and to discover what post-writing procedures (editing, packaging, promoting and marketing), work for you.
Short stories can work with smaller sets of characters, allowing you to focus on the individuals without having to fill up the ranks.
Short stories can each be rendered with a different writing style and voice; allowing you to discover your preference and proficiency.
Comment: Just want to second your point about learning the literary life cycle – going through the revising/redrafting process a few times with short stories really helped to prepare me for the rigour required when writing a novel and to highlight my own strengths and weaknesses on a manageable scale. (This is after I tried the 'discovery-write drivel' approach and found it inadequate. But, as John Cleese once said, 'any drivel can lead to the breakthrough'. I still discovery-write a few scenes as part of the planning process to generate ideas – drivel has its place!)
Answer: How to use short stories to explore the world? I'd say design the world according to the logical incentives that drive the structure of a society. Address the people (workers? Slaves? Castes? Indentured servitude? Free citizens?). Address the leaders (Oligarchs? Chieftans? Elected officials, aristocracy, feudal lords, revolutionary leaders?). Address the scale of the society (tribal, regional, national) etc.
Once the important questions are answered you can use short stories to explore the setting you've built up. Make sure each short story demonstrates something essential about the fictional society to the reader. Then, make sure each short story has no plot contradictions with any of the other short stories.
Once you have enough short stories, you have an established setting with a rich lore. This lore can be drawn from, for your novel.
Answer: This is somewhat a matter of opinion and personal preferences. So for me, I would not write short stories. If you want practice, write scenes. Then leave them alone for about a week (do not look at them). This is enough time for most people to forget all but an outline of what they wrote. Then try to read them as a reader. Once you have read the scene, immediately start figuring out what you think is wrong with it. What is unclear? What is confusing? Where did you go wrong?
If you think nothing is wrong with your first draft, chances are you should not be a writer; every commercially successful writer I have ever heard talk about it (or read what they say about it) thinks their first draft needs work. Stephen King rewrites five or six times. I have rewritten passages more than thirty times.
But don't be discouraged by this; it is training for what you will need to do.
A book is recursive: It has a beginning, middle and end. Each chapter has a beginning, middle and end. Each scene has a beginning, middle and end. Even the sub-parts of a scene, like a conversation, have a beginning, middle and end!
Write scenes for practice. Plan to throw them away or put them aside, they probably cannot be used in the actual book, but it will help you to learn writing. Write a dozen of them, planning to throw them away.
THEN start thinking about your characters, drawing on your scenes. Who is the hero of this story? Who do you like writing about in your scenes?
You need a WHY for this hero, what do they want, so much they would be willing to sacrifice almost anything (or straight-up anything, including their own life) to get it? (e.g. a woman saving her child, a boy avenging his parents, a slave escaping slavery.)
Depending on how you like to write, you do not need the whole story in your head. Stephen King comes up with such situations and just starts writing, he gets so attuned to each character he has each of them do what they must do, and figures the story must resolve itself somehow. He says his stories come from developing characters readers will like, and then "putting them in the cooker" to see what happens next.
To me, what you need is Act I: The setup, the introduction of characters (including the hero and villain or the hero's big ass problem if there is no villain), and the setting and rules: For example you can't introduce magic in Act II (the middle) or Act III (the resolution of the big ass problem), you can't suddenly make your hero a consummate marksman or martial artist or able to speak Chinese if that was not introduced in the setup.
Write a scene. Put is aside, write another. Repeat. come back in a week and figure out what is wrong with it, how it could be better. Rewrite it, put it aside, repeat a few times, until you think you are on par with a pro. All while planning to push it all aside when you begin the novel. Still, this will help to acquaint you with your characters, both the ones you have in mind and the ones you discover are needed while writing these scenes.
Writers learn to write by writing and rewriting and rewriting again. Plus you must have the ability to criticize your own writing (at least after letting it fade somewhat from memory).
You can start with your first work intended for publication being a novel. But I wouldn't start writing a novel until you have some experience writing at least scenes that can span several pages. Practice that first.
Added due to commentary:
There is some question about whether I really mean to write something with the intention of discarding it. Yes, I really mean that. I said "they probably cannot be used in the actual book," but I'd go further: They won't be used in the actual book, or screenplay or whatever.
The way I write, I probably write three pages for every page in the final work. I am used to that. Do the math: A 250 page books means I wrote 750 pages and threw away 500. Now if I know stepping into this mud I am going to throw away 500 pages, why not do some of that intentionally, to save time?
Here is an example. I have a female character in mind; Annie. (I may rename her later, at first I tend to give my characters names starting with A,B,C...). Annie is a 28 year old private pilot, single and unattached. She learned to fly in the Marines, doing six years. She's flown in combat, she's been shot down and ejected, parachuting into the border of enemy territory. She escaped to friendly territory, but shot and killed three male enemy combatants in pursuit, one she shot in the head while wrestling for a gun. From her aircraft, Annie has bombed and killed dozens of people; probably of all ages and genders and more than she can know. Annie remains Marine fit; she works out religiously wherever she is, and continues her fight training two days a week if not away on work. She is average in looks, not a stunner or even close to a model, but attractive in the way that strong and healthy looks sexy. She is a heterosexual.
She isn't suffering from PTSD. She's a patriot but a political cynic, and thinks lives are being wasted. She doesn't hate the Marines. They got her through college, taught her to fly in the worst possible conditions, they made her who she is.
Now the big question (for me) is: Why? How did Annie get to be THIS Annie? When did she know she would join the Marines in war time? When did she first know she wanted to be a pilot? Who inspired her? What are the parents like that produced a girl like this?
As an author, I want this Annie to fall in love. Will it be her first time? It is very rare for a 28 year old to be a virgin; I don't see Annie as one. What was her first consummated romance like? Did she plan to lose her virginity, or was that an impulsive act? What was her partner like? What was Annie like in high school?
I am not a fan of deep back stories; they bore me. But pivotal scenes do not, and writing some of those pivotal scenes in Annie's early life, before she was a Marine, even though I do NOT want them in this book, can help me see who she is, if my sketch needs to be revised, and can even help me find the next character: The man with whom she will fall in love. Perhaps like the father she admires, or the boyfriend she had in high school. Perhaps Annie the Warrior falls in love with Bryce the architect that has never been in as much as a fist fight. I don't know what turns her on, but I can find out by looking for those pivotal life scenes and trying to write them, and discovering the core qualities of a young girl that will become Annie the Warrior, and Annie In Love.
To me the point isn't to include these scenes in the book, they are fragments without a plot and outside the plot I want to write, about an adult Annie. I don't like flashbacks, either. So I'll put them aside. The point is to write a novel that could be a screenplay that is good enough to sell. Not to write a novel fast, not to write a novel from start to finish without a mistake, not to make sure every word I write gets sold. The goal is not efficiency, the goal is good coherent writing people can believe. The spirit of what I write will end up informing my treatment of the Adult Annie. In that sense the time and effort and pages are not wasted at all, I built a "real" girl and we meet her in the middle of her story without fumbling around.
Comment: *"A book is recursive"* Actually, what you describe is not really recursion. Recursion would be if *a book contained a book*. What you describe sounds more like a *fractal* in that the smaller pieces have the same overall shape as the larger piece.
Comment: @MichaelKjörling Fractal works too. But "recursive" means characterized by recurrence or repetition; or the repeated application of a rule (e.g. X factorial equals x times (x-1) factorial). That is how I use it here; the rule is each part of a book has a beginning, middle and end. So each of those have a beginning, middle and end. And so forth, but like all recursive algorithms, there must be a stopping point or we are stuck in an infinite loop. One could argue about whether all sentences have a beginning, middle and end, I suppose, but in writing I'd stop somewhere in the paragraph range.
Comment: I like your suggestion of writing scenes for practice structuring, connecting and balancing scenes means A LOT to the overall project of writing. I completely disagree, however, with your suggestion of writing something that plan on throwing away. At least for me, this would completely remove the motivation for writing. Yes, writers end up writing a lot, and sometimes entire scenes, that will not make the final edit, but most (if not all) of the writing is likely done thinking that it will be part of the story. It might be your phrasing that gets me, but it seems to me that all scenes written-
Comment: - should feel useful and meaningful for both practice AND content. Yes, we get to know our characters more every time we write 'about them', but I bet very few scenes are purposely written, simply for writers to to get to know their characters. In theory it makes sense, but in practice, I doubt many people's creative process allows for this kind of writing-to-later-delete manor. Therefore, instead perhaps write shorter and longer scenes that your characters could refer to, to describe each other - or for you to describe them to someone who doesn't know them.
Comment: @storbror To each his own, I have learned in forty years to not be so greedy or straight-line or "efficient" or reluctant to use an (intellectually) brute force approach to getting the job done. I rewrite, rewrite, and rewrite again anyway. I attended a photography exhibition (NYC) of about 20 pictures. Every one, ***stunning.*** Someone asked the photographer how many pics he took to get these 20: He said these were the best of 12,000. A 600:1 ratio. I love to write. I don't mind writing 600 pages to produce 280 that get published. We just have a different mindset about this.
Comment: I completely agree that far from everything we write will or should necessarily make it to the final edit. I'm just saying that I doesn't currently make sense to me to write anything "knowing it won't be used" - that would remove most to all motivation.
Comment: @storbror Yes, I heard you the first time! IMO it should not, the motivation is to do some work to flesh out how your characters behave. The scene *might* be used, but insisting on this "efficiency" of never writing anything that won't be used is, IMO, irrational and misguided. An artist paints a sunset for a background: Then covers much of it with trees, buildings, people and animals, a kite. Does he regret painting the parts of the sky that are later covered? No. I don't regret writing scenes to clarify my thoughts, knowing they won't fit in the end. Their ***spirit*** enables the product.
Comment: I don't think we actually disagree on anything about the final product - believe our wording lead to some misunderstandings. I +1'ed your answer, but merely wanted to have another perspective regarding being motivated to write scenes.
Answer: For a science-fiction and fantasy writer in particular, there is one very good reason to write short-stories:  There is a market for them.  There are a number of prestigious and well-read media outlets, both print and online, for speculative fiction.  For that reason, the majority of sf writers "serve an apprenticeship" as short story writers, which helps them perfect their craft and build an audience.  Many of the best known science fiction books started out as stories, that were either collected (Asimov's I, Robot) or expanded (Card's Ender's Game).
With that said, short and long fiction are very different forms, and it is possible to excel at one and not the other.  Even so, given your chosen genre, I think it would be worth a try. One possible way to start would be to pick a key incident from your proposed novel, and try to give it its own compressed story arc (beginning, middle and end). Or, pick a backstory from the world you're building, one that might not make it into the book, and bring it to life.  That way, even if it doesn't work on its own, it still contributes to your larger novelistic goals.
Answer: I don't now about you, but when I have written something it is "done" for me and the need to write it again is so low that I would never turn a short story into a novel. Also, a short story is work, and you might want to save your time and energy for your novel.
You might be different, but either way you can do what writing short stories would do for you, without actually writing them:
What I would do is what people do in all walks of life, that is, generate ideas. How you generate story and character ideas will depend on the kind of person you are, and I cannot tell you what will work best for you. What works for me is:

explore the world (mentally): What are its characteristics? What is interesting and fun about it? What aspects of it do I want to feature prominently in my narrative? How do these aspects affect the inhabitants of that world?
explore possible characters: What kinds of characters and relationships am I interested in? How would these people live in the world I have devised? How would the aspects that interest me about my world affect those characters?

Either your exploration should lead you to the one single story that you feel you want to tell, or a couple of possible stories should present themselves. Evaluate them in the common manner, i.e. talk to others about them, research your market, etc.
Answer: Actually, my novels are by and large a collection of "short stories" where each chapter is a typically self contained story or adventure by a consistent set of heroes, which is building up towards the conclusion which takes place in the last few chapters.  This is mostly because much of my writing takes inspiration from semi-serialized TV where the conflict of the day might be related to the main antagonist of a series or might just be an excuse to explore the setting or just a stab at humor.  That said, my genre I best use this in is a superhero/scifi setting, which I feel needs to establish a good rogues gallery, rather than a single hero vs. villain which tends to happen in the movies and books.
Also, with the rise of e-publishing, I love the idea of serialized stories and making the readers come back time and again for payoffs and the books will more be a collection (think watching a new TV show vs. Binge watching on Netflix).
Comment: This has some good content, but you might want to rework it a little, so it reads more as an answer for a general audience, and less as a comment.about your own writing.


2007 Chevy Malibu key

Question: I lost my only key to my car and the dealership keeps telling us different things every time we talk to them. They said they can make a key from the vin number but that it won't start the car and that we would have to get my car towed (across the island at a huge cost to us) to the dealership. Is there a way to get a new key without towing my car?
Answer: The dealership should be able to make the key from the VIN. Since you do not have a registered key already, you will not be able to program it yourself. The vehicle will have to be somewhere it can be reprogrammed with the new key(s). You can call a lock smith to come out and reprogram the car, but chances are it's going to be just as expensive as having it towed (if you can find one which will do the work for you in the first place). If you had a key already, programming it yourself would be a breeze and take you only a few minutes. Your best bet here is to find the lost key, but I would bet this has already proven pointless.
Answer: Once in a while you get lucky. Have them make you a key from your VIN and try it. Some vehicles will start with an unprogrammed key - my '07 Tacoma is one of those vehicles; It'll start even with the grey (non-intelligent, door-only) key.
Try the key. If the car refuses to start with it, you'll have to go to Plan B, towing the car to where they can program the key for it.

2007 Chevy Malibu key


New hidden service want exposure

Question: I just started a new hidden service on tor and I just wanted to ask what's the best way to ensure that people know it exists???
Answer: Your options are the same as for a clearnet site, advertising.
Use (anonymous) social media accounts to advertise the site. If you want to stay anonymous, make sure there is nothing in the social media profile that could lead to your identity (use a new email created over Tor).
Tell people that might be interested in your site about it. Again, if you want to stay anonymous, you must be anonymous to them.
Advertise on other hidden services. If other hidden services have a link to your site, it will be indexed by hidden service search engines.
Answer: Just, try to connect using your Tor browser bundle. If you configured your hidden service right, you should see you server up and running.
If not, then you should check if you torrc is correct. Here is how you do it:

Type tor --verify-config or in Windows tor --verify-config | more
to check your settings.
Check for output for this line: Configuration was valid.
If there is no such line, you should check your torrc file.

Also this webpage might help you.
Comment: So what does that do that ensures people finding out about it???

New hidden service want exposure


Is there a way to get a text percentage for battery capacity?

Question: On other phones I have had, I could get information with respect to the battery drain, like "45% capacity" (as opposed to the visual representation of a two-tone battery). Is that information available on Windows Phone 7?
Answer: I really don't know if I understand but did you mean "Setting"->"Battery Saver" where you can see "battery info"?
Comment: I don't think windows OS allows third party apps to fiddle with top notifications. Applications are just allowed to use live tile notifications. So may be you can look for an app that works In that manner and show you the battery info on tile
Comment: I don't believe the current battery capacity is made available to app developers...
Comment: Yeah, that works! Thanks. But I would prefer being able to double-tap the battery icon to get the info.
Answer: I use Battery Level. It has a live tile that shows your % all the time, and can show on the lock screen if you want. If you tap the tile you get a graph of your battery level over the previous day. You can also get it to tell you the battery level aloud if you want.
On my WP 7.5 I have to go into settings, scroll down to battery, and look at the tiny print. I definitely prefer having the live tile with a text percentage on my WP8.
Answer: There is an app for WP8 called Flashlight-X that will show remaining battery percentage. I pinned it to my start screen (live tile). I hit it & the flashlight turns on via the camera's LED. It also shows battery percentage left.


Closing/Migrating question instructions/guidance

Question:
What does it mean to "close" a question?
How/when can I vote to close questions?
When should I vote to close questions?
When should I not vote to close questions?
Comment: This was originally a feature request for more descriptive close messages, and probably should never have become an FAQ entry. It should be somehow merged with [What is a "closed" question?](http://meta.stackexchange.com/questions/10582/what-is-a-closed-question).
Answer: Close reasons have been discussed before on Meta Stack Overflow, to some effect. And while most of us who are trigger-happy with the close button know what they mean, I think there does exist a general lack of guidance on when they should be applied.

What Does "Closing" a Question Mean, Anyway?
Closing a question is not an inherently negative activity. It should not be considered so, by either those closing, nor the original poster (OP). Instead, "closing" a question should be considered a form of sequestering -- placing the question in a state of Limbo.
It is not only possible, but in the right circumstances, likely, that the question can be reopened. It may just need input from the original poster -- clarification of the question, differentiation between it and the supposed duplicates, etc.
Closing a question merely prevents new answers from being added. The question, and any existing answers, can still be edited, voted and commented upon. This allows activity to flow, which in turn allows the community to determine if the question should remain closed.
If your question is closed, read below and see if -- objectively -- you can understand why it was closed. Over time, you'll likely find that you agree with many of the community's closures, and when your rep allows, you may very well end up joining the cleanup crew.

Here are the rules I personally follow. I have made this post Community Wiki, with the expectation that other question closers will actually edit these rules, and edit out all of the mentions of "my" guidelines. My sincere hope is that, perhaps, this can become something of a "community consensus" guideline.
Without further ado:
Exact Duplicate

This question covers exactly the same ground as earlier questions on this topic; its answers may be merged with another identical question.

In my mind, this means that the questions are, for lack of a better term, identical. I do not believe in deleting duplicates, as they serve an explicit SEO (and human search) benefit. However, if a new question covers absolutely the same ground as an old one (or the combination of two or more, e.g. a question asking for C# books and blogs is a duplicate of two existing questions), then I definitely believe the new one should be closed so that there is one canonical location for the answers to that question.
Additionally, if the answers from another question adequately answer the new one, or any answers posted to the new one are likely to be duplicates of answers from the old one, you're probably looking at a duplicate -- even if it isn't an exact duplicate.
When Should This Not Be Used?
In addition to the obvious, "if it's not a duplicate," I actually think this gets overused on Meta Stack Overflow quite a bit with similar questions. Similar or related questions are not necessarily duplicate questions. For example, the question this precise answer relates to could have been closed as a "duplicate" of 5018, but thankfully (thus far) wasn't.
Before rushing to a "duplicate" judgment, ask yourself if there is something about the new version which makes it unique, different from the old. Can you envision possible answers that would apply to the new one and not the old? If so, then its answers likely will have value separate from the old question; it is not a duplicate, and refrain from the closure vote.
Off-Topic

Questions on {site} are expected to generally relate to {topic}. This question is very far afield from {topic}.

This should be self-explanatory, but for many people, it is for some reason confusing. We'll use SO, and programming, as our example.
Programming-related questions are those specifically related to code. Questions which relate to being a working programmer ("as a programmer, how do you deal with a non-technical boss," etc.) are not programming-related. One could easily substitute the words "accountant" and "non-mathematical" for "programmer" and "non-technical" and get essentially the same responses.
Programmer-related is not the same as programming-related. If it's not about code, it could very well be a candidate for closure under this reason.
When Should This Not Be Used?
Gray areas. If you have enough reasonable doubt that it could be on-topic, then leave it alone. If it should be closed, others with more experience will handle it for you. For example, issues with IDEs or questions on the use of source control are not strictly programming-related, but the people most qualified to answer will be on Stack Overflow, as these tools are used primarily in the practice of programming. So leave them be.
Subjective and Argumentative

It's impossible to objectively answer this question, and the question was asked in a confrontational, argumentative way.

Everybody's favorite. It's been the subject of many a battle, and even some advice on how to game the system. Here's the thing: Note the word "and" in the title. In technology, many things are subjective. The way you approach problem x may not be the way you approach problem y, even if they're remarkably similar, due to a variety of potential constraints that may not be verbalized in the question. Also, certain subjective questions are not only on-topic, but actually valuable to the community -- justifications of languages, platforms, operating systems, etc., are all (in my mind) actually valuable to their respective communities -- done once. It's a good thing to have a canonical location of pros and cons.
So when should this be used? When the tone of the question is in fact both subjective and argumentative. "Tell me why PHP doesn't suck" would qualify. "List the pros and cons of using PHP" would not. In many cases, editing out the OP's argumentative stance may salvage the question -- however, once that is done, a careful duplicate check should also be endeavored, as chances are good that at least one site in the Trilogy has answered the question at that point. Caveat: some arguments are unavoidable, no matter how politely the question is asked; if the question boils down to Tabs vs. Spaces, Emacs vs. Vim, Linux vs. Windows, Mac vs. PC, etc... Then leaving it open is just an open invitation for another skirmish in these endless wars. Either find a way to change the focus of the question, or vote to close.
When Should This Not Be Used?
When the question is only subjective, but may hold value to the community, and is not a duplicate of an existing question. On Stack Overflow and Super User, I personally believe this isn't likely to happen until some more brand-new technologies come out.
Not a Real Question

It's difficult to tell what is being asked here. This question is ambiguous and vague, and cannot be answered in its current form.

My personal favorite. I wish it was available on Meta Stack Overflow, to be honest. Many people probably consider this merely a "catch-all," but I don't. Sometimes, an asker merely needs a bit of motivation to finish his thought, and closing can be that "stick". To me, this should be used when one or all of the following conditions are met:

There really isn't a question hidden in the post. ("Windows sucks!" is not a real question.)
The question is so ambiguous that the OP's intent cannot logically and reasonably be inferred. A certain recent question on tagging fell into this category for me; I was its only close vote. (Even though Meta Stack Overflow does not have "not a real question," I voted to close anyway because of this bullet point.) To me, the Golden Rule of editing is to never change the OP's intent. If the question is worded so vaguely that intent cannot be inferred with 100% certainty (for example, there are multiple possible, even likely, intents), you cannot edit it in good conscience -- so vote to close as "not a real question" instead. In this scenario, it's a good idea to comment and ask for clarification from the OP as well. If that had been done in the early stages of a certain recent controversy, said controversy might have been avoided. (Even though I still agree that closure was warranted there.)
(more bullets to come from the community?)

When Should This Not Be Used?
Greg Hewgill says it quite effectively in his answer to this post: "Just because the answer is "No" doesn't mean it's not a real question." In fact, if there is an answer, and the question is unambiguous, then guess what -- it's a real question.
Simple/trivial questions are real questions.
Questions you consider boring are real questions.
Questions you can answer with a 2-second Google query are real questions.
Too Localized

This question would only be relevant to a very small geographic minority, and is not generally applicable to the worldwide audience of the internet.

I'll be the first to admit that I don't hew to the exact definition above. I consider "too localized" to not only be geographically centered, but also to an extent, situationally centered. The obvious closures here would include things like, "where can I find a user group on x in town y." Less obvious choices would include discussions of laws and regulations, distance working, etc. (However, an equal argument can be made in many of those that they are not {topic} related.)
When Should This Not Be Used?
Don't blindly slap this on anything that happens to be location-related. A question on where to buy replacement power cords in the south of France might be too localized, but a question on power connectors used in France wouldn't have to be (for Super User at least). Similarly, asking for help translating your program's UI into es-mx is crossing the line, but asking for help designing your UI to support localization because you plan to sell a version in Mexico isn't.
Don't use this to lock out obscure topics that would otherwise be on-topic for the site. You may not care about operator precedence in Crystal Reports macros, but someone does and sooner or later they'll get an answer (may God have mercy on their souls).
Belongs On {site}
This should be used when a question is clearly intended for a different site in the Trilogy. This should only be used when you can clearly identify exactly which site the question should be on.
For example:

Configuration of a client on a Windows domain will likely belong on Server Fault.
Installing Windows on a PC will likely belong on Super User.
Developing a tool to aggregate information from various servers via WMI and ASDI will likely belong on Stack Overflow
A question about why one of the above belongs on its appropriate site clearly belongs on Meta Stack Overflow.

As you can see, there is some ambiguity in most of the above samples, but they all have a fairly clear home, based on the missions of the site.
When Should This Not Be Used
I'm with Jeff Atwood: When in doubt, don't migrate. Unlike other closures, migration is damned difficult to undo. Once migrated, the original question is locked, which takes closure to a new extreme by preventing commenting. Additionally, there is the possibility (actually, likelihood) that the OP does not have an account on the migrated-to site.
This should only be used when you are 100% certain that not only does the question not belong on the site where it was asked, but that it absolutely does belong on the site you are targeting with the migration.
NONE OF THE TRILOGY SITES (including Meta Stack Overflow!) ARE DUMPING GROUNDS!
Noise or Pointless

This question does not add anything useful; having it present on the site is actively harmful because it distracts from other more useful questions.

This is a Meta-only close reason, and is roughly equivalent with "too localized," "argumentative," "not a real question" and the old closing reason of "blatantly offensive." It's difficult to adequately define "noise or pointless," but it's much like pornography: You know it when you see it.
Typical examples would be full-on rants and ravings, ludicrous joke questions and spam or offensive material.
When Should This Not Be Used
When the question is valid or legitimate, but merely controversial. Also, if the question is severely inflammatory in its current incarnation, but can be salvaged with editing into a reasonable (if potentially controversial) post, then edit before closing.

So there we have it. A starting guide. I now open the floor to all comers. If you're about to comment and disagree, but have closure rights on one of the Trilogy sites, I implore you to edit this as well as comment. Change what I've written -- that's why this post was CW. I'm trying to get us to a decent consensus on this.
As C. Ross commented on my "real" answer to this question, not all of those with closure rights will read this. In fact, I suspect most won't. And as Roboto mentioned in his comment on the same answer, we are fallible. But if those of us who are highly active in the edit/closure space do, and come up with a set of good guidelines, I think it will serve to benefit the community over time as we get our houses cleaned up.
Remember: Closing isn't negative, but those of us with closure power must use caution when we exercise it -- when we are just with our decisions, there is no controversy. When we are unjust and unnecessarily obstinate, the community will rebel against us.
Comment: Nice work, Smurf 'bu. Edited in a few extra thoughts...
Comment: Great list.  People think all of the reasons are confusing, but I like being able to choose the right reason to close a question.
Comment: If it makes you feel better, you got a second cigar-smoke LOL on God have mercy on Crystal Reports macro operator precedence users' souls. (My lung specialist thanks you, BTW, his kids need this college fund.)
Comment: This is severely outdated. Are you considering updating it? [/faq-cleanup]
Comment: CAPTCHA! "cided blazers"
Comment: I LOLed at Lotus Notes. Did you put that in there just to make sure we were paying attention? (BTW, you owe me my next doctor copayment; LOLing while actively puffing a cigar and swallowing a swig of bourbon is hazardous to your health!)
Comment: To answer your question: yes. To commiserate with your reaction: I know. :-(
Comment: Excellent answer.  I added a couple of minor points, but you definitely hit all the high notes.
Comment: Some of these reasons may no longer exist.
Comment: @random: Excised Spam, Blatantly Offensive and No Longer Relevant. I haven't been active much since Friday (my time), and didn't know about the closure changes.
Comment: The change was only a couple of hours ago really. So not that long.
Comment: I'm thinking I might need to add some stuff from "no longer relevant" to "too localized," as it seems that's where that's supposed to be now ...
Comment: I don't think this is too outdated, specially when given the context of its age... like the Joel Test should include something specific about CI/CD. Not to name names, but I've encountered overly successful reviewers using their "power" to gang up on topics they don't like... think Gaming SO, but you don't like anything to do with the Steam Deck...
Answer: Maybe we do need instruction. Just today I've noticed quite a number of posts that are at 4 close votes yet (in my opinion) don't deserve to be closed:

legitimate (if poorly posed) question about the difference between semaphores and monitors https://stackoverflow.com/questions/1873646/semaphores-and-monitors (oh hey it was improved, thanks Shog9!)
I don't think this one needs to be migrated anywhere: https://stackoverflow.com/questions/1872998/does-pipe-have-to-write-temporary-file
Just because the answer is "No" doesn't mean it's not a real question: https://stackoverflow.com/questions/1871350/is-there-a-way-i-can-disable-copy-paste-and-printing-of-the-pdf-file-generated-us
Regarding code licensing, it's not a technical question but I think it's certainly programming related: Searching for non-commercial license for source code
Also definitely programming related, it's a question that programmers everywhere struggle with: Should you include a much-requested feature that is fundamentally wrong?
This one has at least three embedded questions, but every one of them is valid in its own right: https://stackoverflow.com/questions/1864875/how-can-i-convert-java-python-and-c-code-into-standalone-executables
Missing some critical information about platform, but otherwise a perfectly reasonable question: How do I present an open folder selection dialog in Perl?

I expect that some of these will quickly get the one more close vote they need to be closed. I suppose at that point, the close user names are published so we'll know who needs some remedial education on close etiquette. :)
Comment: BTW, this seems as good a time as any to mention this: Shog9, thanks for the rewrites. Seriously. I wouldn't have voted to close the monitor question (it was valid in its original form, but just badly worded), but your rewrite made it crystal clear and a much better question.
Comment: I upvoted you because I agree with most of what you say. However, I disagree with licensing and much-requested-but-wrong-features. Programmer- and project-related are not programm*ing* related.
Comment: I was really on the fence with that PHP/PDF question... In the end, I voted to close because, while *valid*, "no you can't" isn't all that *useful*. If he could provide a bit more information regarding his situation (as the lone commenter requested), it might collect some useful answers - and I suspect that title is gonna catch quite a few Google hits.
Comment: These things are exactly the type of things I think guidance on would be great.  Obviously after all is said and done, this will still be a touchy issue for some questions.
Comment: @Shog9:  I voted to reopen that one.  It is a legitimate question.  It's answerable ("No."), and one that other people might be interested in.  On the other hand, I'd already voted to close the project-related one.  I probably interpret "programming-related" wider than most, but I just couldn't see why that wouldn't apply to anybody doing, say, workflow routines that didn't involve computers.
Answer: OK, a couple of notes:

There actually is guidance there. There are tooltips on every reason. However, it's not detailed guidance, I'll grant that.
There actually is a detailed discussion here on Meta that goes over all the close reasons and what they actually mean, why they should be used, etc.
"When in doubt, don't" refers specifically to migration. I agree with that for migration, because that's a very big step to take. However, for non-migratory swallows answers, closure is not necessarily bad (if done for the correct reasons).

Perhaps there ought to be some specific guidance about each close reason. I'm thinking this question right here could provide those answers. When I get home this evening and have more time, I'll post a CW answer here with the guidelines I try to follow. (It'll be CW not to avoid downvotes, but so that as a community we can get to a decent set of guidelines.)
If someone else wants to start that process, then I say go for it, and I'll just pile on to that answer.
Comment: "Closers are still fallible." Damn right we are -- we're human, we make mistakes just like anyone else! I grant that I've voted to close when perhaps I shouldn't have. The scenario you describe is exactly why we now vote to close (it takes 5) instead of having the power to do it immediately. Other closers may not agree with the vote system, but I personally do. It allows a basic sanity check. And the ability to reopen provides the counterbalance to it.
Comment: I agree with your points.  The thing is, how many of those 2000 odd users will check on Meta.  Probably still worth doing.
Comment: If it gets in the FAQ ... You might be surprised.
Comment: I can tell you that despite the tooltips, the closers are still fallible. I posed a question here: http://stackoverflow.com/questions/1798552/staying-relevant-as-a-programmer and it was initially closed as "not programming related" - What?! Even Shog9 said that was a bad close reason. I managed to get it re-opened which garnered more views, votes, and someone got extra rep because I posted a bounty. The moral of the story is if it is popular enough, it will be re-opened against any of the closers will. In my case, the closing was pre-emptive and you can see I didn't really alter the question.


How do I get my Bitcoin Core to rescan for a different wallet.dat file on OSX?

Question: I have an old wallet.dat file that I have backed up and I have already replaced the new wallet.dat, which had no btc on it with the old one, which does. I have tried to enter the command for the rescan function to adapt to the different wallet information without success. Do I do this through the QT terminal or through the debug console command line? And can this be done before the bitcoin core wallet is completely synced with the entire network? Thanks.
Answer: The QT terminal (RPC) rescan option will be available in 0.16 (not release by the time of writing). https://github.com/bitcoin/bitcoin/pull/7061
Manual rescans should generally not be required, though, if you are missing transactions in an old wallet, the rescan may help.
In Core < 0.16 you need to rescan either by startup argument or by a configuration option.
Startup (OSX)
Open up "Terminal.app" and enter /Applications/Bitcoin-Qt.app/Contents/MacOS/Bitcoin-Qt --rescan
bitcoin.conf (OSX)
Place a plain text file called bitcoin.conf in ~/Library/Application Support/Bitcoin/ and make sure the textfile contains the line "rescan=1".
Make sure you remove the rescan=1 line after you have successfully rescanned your wallet.dat.


Deletion of SQL Profiler Trace files (.trc)

Question: We've noticed a lot of .trc files in our SQL data folder (\Program Files\Microsoft SQL Server\MSSQL.1\MSSQL\Data) on our server. The date range for these files spans over one day and the total file size of all files together is about 21 gigs. I'd like to free up this space but I'm not sure if I can just delete the files manually through Windows Explorer or if I need to do anything in SQL, like run a command or script. Any ideas?
Answer: The .trc files are safe to delete.
.trc files generated by SQL Server in process of saving events to a physical file without using the Profiler client tool.  Server-side tracing is enabled and controlled by using SQL Server system-supplied stored procedures and functions. With these system-supplied processes, you can identify what to trace, when to start and stop tracing, what traces are running, and view trace information stored in the trace file.
View the number of currently running traces :
SELECT count(*) FROM :: fn_trace_getinfo(default) WHERE property = 5 and value = 1

More detail about the running traces:
SELECT * FROM :: fn_trace_getinfo(default)

You can terminate a trace with the 'sp_trace_setstatus' stored procedure using the traceid:
EXEC sp_trace_setstatus 1, @status = 0
EXEC sp_trace_setstatus 1, @status = 2

Setting the status to 0 stops the trace
Setting the status to 2 closes the trace and deletes its definition from the server
Good Luck!
Answer: If the trace files are no longer in use, then they should be fine to delete. I'd say someone was doing some troubleshooting of an application and forgot to delete the trace files when they were done.
A trace file would only be in use if the trace is still running. If the files are more than a day old then it's a safe bet that the trace is no longer running. If the trace IS running the file will be locked and you won't be able to delete it anyway.
Answer: are the traces still rolling over? what are the files name like? make sure it's not something like an audit trace.  Filenames are typically audittrace[datetime].trc
It could also be the default trace (2005+), although its unusual that it would be that large in size and these are typically found in the \MSSQL.1\MSSQL\LOG
Just to be sure, have a look at sys.traces to make sure the trace isn't still running , or stopped but still defined, or it is in fact the default trace.  You should only be seeing your default trace in there if its turned on, with an id of 1.
Either way, as long as you don't need the traces their safe to delete.  You just want to be sure they dont come back!


Kenmore Dryer gets warm but not enough to dry clothes

Question: My Kenmore dryer heats up a certain extent but not enough to dry the clothes in the cycle.
Comment: Gas or electric?
Comment: How old is the dryer?  When is the last time you cleaned the lint from the dryer (not just from the screen)? How is the dryer vented? When is the last time the vent line was cleaned?
Comment: Okay, but what's your question?
Answer: A dryer works by forcing air past a heating element, through the drum where the clothes are, and out a vent. If there are any restrictions in the air flow, not enough hot air will move through the system.
Let's take this dryer for example

The motor rotates the drum, which tumbles the cloths. It also spins the blower, which draws air in through the intake, through a heater (which could be gas, electric, etc.), through the clothes filled drum, past the lint screen, and eventually out the exhaust.

Notice after the air leaves the drum, it takes moisture and lint with it.  Some of the lint is caught by the lint screen, but not all of it.  Over time, lint can build up inside the dryer beyond the screen. If this lint is not cleared out, and is allowed to accumulate. The air flow through the entire system will be restricted.  Which means less hot air will move through the drum, and your clothes will not dry.

Another common problem, is flexible exhaust tubing.

This stuff doesn't have a smooth interior wall, which can often lead to lint accumulating in the pipe. This also leads to a restriction of air flow through the system, and again reduced drying.

Often times this flexible tubing is plastic, which is even worse since it's flammable. If for some reason the lint in the tube ignites, the tube itself will burn as well rather than containing the fire.

tl;dr
The easiest ways to insure proper drying are:

Clean lint screen after every use.
Remove lint from inside the dryer (see manufacturer documentation for schedule and procedure).
Remove lint from exhaust plumbing.


Over the long term, why invest in bonds?

Question: Over the long term, stocks have historically beaten bond returns, even after accounting for the periodic market crashes. The general consensus is that your portfolio should at least be 25% in bonds. If I don't need this money for decades, meaning I can ride out periodical market crashes, why would I invest in bonds instead of funds that track broad stock market indexes?
Comment: To reduce your overall returns.
Comment: Current bond yields are lower than the rate of inflation.  Buying them would lose you money.  If you don't want to be fully in the market right now I would suggest paying off debts instead.
Answer:
If I don't need this money for decades, meaning I can ride out periodical market crashes, why would I invest in bonds instead of funds that track broad stock market indexes?

You wouldn't.  But you can never be 100% sure that you really won't need the money for decades.  Also, even if you don't need it for decades, you can never be 100% certain that the market will not be way down at the time, decades in the future, when you do need the money.  The amount of your portfolio you allocate to bonds (relative to stocks) can be seen as a measure of your desire to guard against that uncertainty.
I don't think it's accurate to say that "the general consensus is that your portfolio should at least be 25% in bonds".  For a young investor with high risk tolerance, many would recommend less than that.  For instance, this page from T. Rowe Price suggests no more than 10% bonds for those in their 20s or 30s.
Basically you would put money into bonds rather than stocks to reduce the volatility of your portfolio.  If you care only about maximizing return and don't care about volatility, then you don't have to invest in bonds.  But you probably actually do care about volatility, even if you don't think you do.  You might not care enough to put 25% in bonds, but you might care enough to put 10% in bonds.
Answer: Bonds provide protections against stock market crashes, diversity and returns as the other posters have said but the primary reason to invest in bonds is to receive relatively guaranteed income. By that I mean you receive regular payments as long as the debtor doesn't go bankrupt and stop paying. Even when this happens, bondholders are the first in line to get paid from the sale of the business's assets. This also makes them less risky. Stocks don't guarantee income and shareholders are last in line to get paid. When a stock goes to zero, you lose everything, where as a bondholder will get some face value redemption to the notes issue price and still keep all the previous income payments. In addition, you can use your bond income to buy more shares of stock and increase your gains there.
Answer: I can think of a few reasons for this.
First, bonds are not as correlated with the stock market so having some in your portfolio will reduce volatility by a bit. This is nice because it makes you panic less about the value changes in your portfolio when the stock market is acting up, and I'm sure that fund managers would rather you make less money consistently then more money in a more volatile way.
Secondly, you never know when you might need that money, and since stock market crashes tend to be correlated with people losing their jobs, it would be really unfortunate to have to sell off stocks when they are under-priced due to market shenanigans. The bond portion of your portfolio would be more likely to be stable and easier to sell to help you get through a rough patch.
I have some investment money I don't plan to touch for 20 years and I have the bond portion set to 5-10% since I might as well go for a "high growth" position, but if you're more conservative, and might make withdrawals, it's better to have more in bonds... I definitely will switch over more into bonds when I get ready to retire-- I'd rather have slow consistent payments for my retirement than lose a lot in an unexpected crash at a bad time!
Answer: Many folks use bonds to diversify their portfolio since bonds rise and fall in value at different times and for different reasons than stocks.
Bonds pay interest on a regular basis (usually monthly or quarterly) and so some people invest in bonds in order to match the interest payments to some regular expense they might have. The interest payment does not change (fixed income).
For individual bonds, there is a maturity date at which you can expect to receive the face value of the bond (the issuer's creditworthiness is important here). You can make a little money on a bond by buying it when its value is lower than its face value and either selling later for a higher value, or waiting for it to mature.
Often the minimum investment for a single bond is high, so if you don't have a large enough amount, you can still get the performance of bonds through a bond fund. These do not mature, so you don't have a guarantee of a return of your investment. However, they have access to more bonds than retail investors, so the funds can keep your money more fully invested. If you don't need the income, you can reinvest the dividends and have a little extra capital growth this way.
Comment: Sorry, I tend to ramble.... If you don't need the money for decades, and you're more interested in capital growth, you probably don't want bonds in your portfolio.


Is it possible to select particular portion of a video in adobe premiere pro?,

Question: Im a beginner to video editing, kindly help me to select (free select, as like in Adobe Photoshop) particular portion of a video, to merge with another video?
Comment: Joy, it is not clear what you are asking.  What are you trying to accomplish?  Are you talking about wanting to select some portion of a video to copy from one video to another while ignoring the rest?  If so, how do you want to track it from one frame to another.  Is it stationary or something that moves around?  If this is what you are talking about, it is not easy in most cases.
Comment: Yeah..@Henderson you have understood my query very exactly.. yes i need to select some portion of the stationary video., and also suggest me some tips for motion video also.
Comment: One of the best software to change a selected area in a video clip is Imagineers mocha.Using adob after effects it can creat a new video file with the help of mocha.
Answer: If the element you need to put on the video doesn't move and the shot you are putting it in doesn't move, the best bet is to actually use Photoshop to extract the object you want to place in another video and put it in a file format that supports a transparent background (TGA, PSD, etc.)  You can then import it to Premiere and place the image wherever you want in the frame using the Motion effects settings to define the size and placement.
If the video you are placing it in is not stationary, then it will be necessary to perform motion tracking on the video you want to place the image within.  This is much more complicated and is generally done in something like After Effects.
Comment: It sounds like Joy might be looking for just a basic overview of video compositing? It's hard to tell how simple or advanced the request is.
Comment: @ZeroFlux - yeah, agreed.  That's why I was trying to flush it out a bit more.  I decided to limit my answer to what can be done easily in Premiere and what seems to match with the simple situation he described, but I think your answer adds value too for the more complex case.
Answer: To build on AJ's answer in a different direction, you may want to look at After Effects, which is a popular compositing tool. In After Effects, you can use the pen tool (somewhat like the lasso tool) to create a custom shape. With that shape, you can create a mask for the video. This mask can then show only the video inside of the shape.
Further, the mask can change over time with your video. If the subject of your first video moves, you can adjust the mask's shape as well. In other words, the mask can be animated, though this process can take a lot of time.
Answer: Unless I'm misunderstanding your question, I would try a Garbage Matte effect (either 4, 8, or 16-point).  Stack the two video clips on the timeline (e.g. one video clip on track V1 and the other on V2).  Drag n Drop the Garbage Matte effect (8-point is a good starting point) on to the top video clip.  Activate the Garbage Matte effect in the Effect Controls window and then drag the points to select the portion of the top clip video that you want.
Only the part of the top clips video that lies within the bounds of garbage matte markers will show through.  The rest of the video will be the bottom video clip.
This is reasonable video on the subject. He basically merges two video clips of the same scene with different exposure levels to get a single video with good exposure throughout:

Search youtube for Garbage Matte premiere and you will find several helpful videos on it.


Jmeter doesn't make the request to Server

Question: We are using Jmeter 2.11 for doing performance testing. We set up test with 1000 users with ramp up time as 4 hours. We expected the run will go for 8 hours. But server didn't get request after 3 hours. I'm not sure whether Jmeter not able to make a request or server issue
Answer: The main purpose of load/performance testing is finding limits or bottlenecks on the integrated stack of the application. When you run a load test you should monitor the application/servers closely to understand what is happening. Often these tests are performed with the help of the system administration team, because they have better experience in monitoring the server-side of the systems.
You want to understand which part of the stack cannot handle the load. Server IO, Database access, network, etc... Without adequate monitoring you will keep guessing and making assumptions.
I think you can use the JMeter Server Agent to collect data on the server performance and generate metrics from it. Otherwise monitor the systems in realtime and check the logs for any issue's.
I expect you overloaded the server and it cannot handle 1000 simultaneous users. Of-course JMeter itself could be the problem, but you do not give enough information to diagnose. Maybe you can start JMeter on multiple systems with less users. Instead of ramping one machine to 1000, try four machines that ramp up to 250 users instead.
Answer: Check jmeter.log and logs of your application server for any error messages if they are - update your request as it would be more clear what's going on.
Given you have powerful hardware on load generator (JMeter) instance it can spawn 1000 concurrent threads, but make sure that you're following JMeter Performance and Tuning Tips advice.
If single JMeter instance cannot create required load you can consider Remote Testing when single Master orchestrates load of multiple client load engines.


The sum of the coefficients of $x^3$ in $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$

Question: I know how to solve such questions when it's like $(x+y)^n$ but I'm not sure about this one:

In $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$, What's the sum of the
  coefficients of $x^3$?
Comment: Why don't you treat $(1-1/\sqrt{8})$ as $y$ and $-x/2$ as $x$?
Comment: @Eelvex: I made a mistake while typing the question, sorry.
Answer: $$
\begin{align}
\left(\frac1{\sqrt{x}}+1-\frac12x\right)^8
&=\sum_{k=0}^8\binom{8}{k}\left(\frac1{\sqrt{x}}\right)^k\left(1-\frac12x\right)^{8-k}\\
&=\color{#00A000}{\sum_{k=0}^4\binom{8}{2k}\left(\frac1x\right)^k\left(1-\frac12x\right)^{8-2k}}\\
&+\color{#C00000}{\frac1{\sqrt{x}}\sum_{k=0}^3\binom{8}{2k+1}\left(\frac1x\right)^k\left(1-\frac12x\right)^{7-2k}}\tag{1}
\end{align}
$$
The terms from the red part of $(1)$ will all have fractional powers of $x$ in them, so we only need to consider the green part of $(1)$. Furthermore, for each $k$, the highest power of $x$ in the green part is $8-3k$. Therefore, we only need to consider $k=0$ and $k=1$.
For $k=0$, the term in the green sum is $\left(1-\frac12x\right)^8$ and the coefficient of $x^3$ there is $\binom{8}{3}\left(-\frac12\right)^3$.
For $k=1$, the term in the green sum is $\binom{8}{2}\frac1x\left(1-\frac12x\right)^6$ and the coefficient of $x^3$ there is $\binom{8}{2}$ times the coefficient of $x^4$ in $\left(1-\frac12x\right)^6$, which is $\binom{8}{2}\binom{6}{4}\left(-\frac12\right)^4$.
Therefore, the coefficient of $x^3$ in $\left(\frac1{\sqrt{x}}+1-\frac12x\right)^8$ is $\binom{8}{3}\left(-\frac12\right)^3+\binom{8}{2}\binom{6}{4}\left(-\frac12\right)^4=\frac{77}{4}$.
Comment: Now that I look closer, the idea is pretty close to Brian Scott's answer.
Answer: The formula you're looking for is
$$
(x+y+z)^8 = \sum_{i+j+k = 8} \begin{pmatrix} 8 \\\ i,j,k \end{pmatrix} x^i y^j z^k
$$
with
$$
\begin{pmatrix} 8 \\\ i,j,k \end{pmatrix} = \frac{8!}{i!j!k!}.
$$
This is known as the multinomial expansion (it works for more than $3$ variables too, you just have to add more indices and modify the multinomial coefficient accordingly). Using this, then you find when does $x^3$ appear in the expansion
$$
\left( 1 + \left( \frac {-x}2 \right) + \frac 1{\sqrt x} \right)^8 = \sum_{i+j+k = 8} \begin{pmatrix} 8 \\\ i,j,k \end{pmatrix} (-x/2)^j (1/\sqrt{x})^k
$$
In this case we must have $j-(k/2) = 3$, which means $2j-k = 6$, and $0 \le i,j,k \le 8$, or we can rewrite this as $0 \le j \le 8$, $k = 2j-6$ and $i = 8-j-k$. This leaves the cases $(5,3,0)$ and $(2,4,2)$. Computing, the coefficient in front of $x^3$ is
$$
 \begin{pmatrix} 8 \\\ 5,3,0 \end{pmatrix} (-1/2)^3 +  \begin{pmatrix} 8 \\\ 2,4,2 \end{pmatrix} (-1/2)^4 = 77/4.
$$
(I got the $77/4$ using WolframAlpha and/or a calculator, no magic there.)
Hope that helps,
Comment: Thank you Patrick, I accepted Brian's answer as it made it clearer. +1 anyway.
Answer: My answer does not differ in substance from the others, but I thought that you might like one that doesn't appeal explicitly to multinomials.
You can avoid fractional exponents by substituting $y=\sqrt x$ and asking for the coefficient of $y^6$ in $$\left(1-\frac{y^2}2+\frac1y\right)^8\;;$$ this is thoroughly unnecessary, but it makes the typing a little easier, so I'm going to do it.
The terms of this power are all of the form $$1^i\left(-\frac{y^2}2\right)^j\left(\frac1y\right)^k\;,\tag{1}$$ where $i+j+k=8$. The exponent on $y$ in $(1)$ is $2j-k$, so you want the terms in which $2j-k=6$. Clearly these can occur only with even $k$. If $k=0$, $j=3$ and $i=5$; if $k=2$, $j=4$ and $i=2$; if $k\ge 4$, $j\ge 5$ and $j+k>8$, which is impossible. The only possibilities, then, are terms of the forms $$1^5\left(-\frac{y^2}2\right)^3\left(\frac1y\right)^0=-\frac18y^6\tag{2}$$ and $$1^2\left(-\frac{y^2}2\right)^4\left(\frac1y\right)^2=\frac1{16}y^6\;.\tag{3}$$ It only remains to determine how many times each of these terms appears in the expansion. For $(2)$ there are $\binom83=56$ ways to choose from which three of the eight factors the $-\frac{y^2}2$ is taken, and once that's decided, everything else is determined. For $(3)$ there are $\binom84=70$ ways to choose from which four factors the $-\frac{y^2}2$ is taken, and for each of those there are $\binom42=6$ ways to decide from which of the remaining two factors the $1$ is taken. The total coefficient of $y^6$ is therefore $$56\left(-\frac18\right)+70\cdot 6\left(\frac1{16}\right)=\frac{210-56}8=\frac{77}4$$ (assuming that I've not loused up the arithmetic at some point).
Comment: That was perfect, thank you.
Answer: The multinomial expansion gives $\mathbf{S}= (1 - \frac{x}{2} + \frac{1}{\sqrt{x}})^8 = \sum_{k_1+k_2 +k_3=8} \binom{8}{k_1,k_2,k_3} (1)^{k_1}(\frac{-x}{2})^{k_2} (\frac{1}{\sqrt{x}})^{k_3}.$ Simplifying we get $\mathbf{S}=\sum_{k_1+k_2 +k_3=8} \binom{8}{k_1,k_2,k_3} (\frac{-1}{2})^{k_2} (x)^{k_2 -k_3/2}.$ Now identify the tuples for which $(k_1,k_2,k_3) \in \mathbb{Z}_{+}^{3}$ such that $k_1+k_2+k_3=8$ and $k_2-k_3/2=3.$ Then add the coefficients for these tuples to get your answer.
Answer: You can just multiply it out.  Alternatively, you can reason the terms are of the form $1^a(\frac x2)^b(\frac 1{\sqrt x})^c$ with $a+b+c=8, b-\frac c2=3$.  Then $c=2b-6$, so $a+3b=14$ and $a$ needs to be $2 \text{ or } 5$.  Then you need the multinomial coefficient as stated by Suresh.
Comment: If $a = 8$ you don't have a coefficient of $x^3$...
Comment: @PatrickDaSilva: You do if b=2, c=-2.  But that isn't part of the product, you are right.  Fixed.
Comment: Things happen. Heh =)


Is there a branch of linguistics using Calculus as a mathematical tool?

Question: Is there a branch of linguistics using Calculus as a mathematical tool? I mean, can we use differential or integrate in linguistic study?
Comment: Calculus helps in continuous phenomenon , discrete calculus is used in  statistical natural language modelling for text documents which involves summations and discrete probabilistic distributions.
Comment: there is lambda calculus for linguistics too
Comment: @Ali Thanks for the reply,I am just wondering if we could introduce calculus into linguistic,that would be amazing.
Comment: http://en.wikipedia.org/wiki/Lambda_calculus
Comment: @AveMaleficum Hello and welcome to Linguistics. :) Since you're new here, check our [FAQ] and [About](http://linguistics.stackexchange.com/about) pages. :D Your question here is good by the way.
Comment: I believe the OP is talking about differential and integral calculus, not (for instance) [first-order quantified predicate calculus](http://www.umich.edu/~jlawler/logicguide.pdf), nor the calculus of errors. The term is very general, and refers to any well-constructed symbolic systems of proof.
Answer: Calculus can be a useful tool in quantitative linguistics. One simple example would be the deduction of the theoretical equation for the Piotrowski law modeling language change by Altmann et al.
The increase of new forms p' is proportional to the product of the proportions p of new and 1-p old forms: p' ∝ p(1 - p). Introducing the factor of proportionality b then yields the differential equation p' = bp(1-p), which can be solved for p as a function of time: p = 1/(1 + a exp(-bt)). (Skipping a few details here, like Piotrowski's earlier fitting of a tangens hyperbolicus curve, which is compatible with the above, since 1/(1 + a exp(-bt)) = 1/2 tanh w(t-t₁) + 1/2, where 1/2 tanh w(t-t₁) was one of Piotrowski's original proposals. Also, this can be (and was) generalized for partial change, and there are other viable approaches besides the interactionistic model.)
If you are interested in quantitative linguistics, I'd suggest the works of Prof. Gabriel Altmann and Prof. Reinhard Köhler.
Comment: Source: Best, Karl-Heinz (Hrsg.) ; Kohlhase, Jörg (Hrsg.):  Exakte Sprachwandelforschung. Göttingen : Edition Herodot, 1983.
Answer: Definitely. This is a little obscure, probably to obscure to be of any real use, but there is a chapter from the book Aspects of Automatic Text Analysis, The Mathematics of Semantic Spaces (PDF), that models semantics as a metric space. It's general calculus on metric spaces, and this is probably the limit of mathematical abstraction in applied linguistics that I've seen. Unless you've been introduced to some very high level math, this material won't make any sense.
Other then that, I know acoustic phonetics has derivatives, integrals, limits, series, etc. For example, one way to describe a sound wave quantitatively is take it's Fourier transform which  involves the evaluation of an integral. Then there's statistical analysis of corpora, where calculus isn't directly involved, but the formulas won't make any sense unless your familiar with the concepts of basic calculus. Things like weighting, infinite sums, neighbourhoods, continuity, finding volumes and lengths, optimization etc. are implicit in a lot of statistical measurements. Including vector calculus. A lot of linguistic models employ continuous sets, not just discrete sets, and require a metric or a topology to define the set as a mathematical space.
Answer: Lambda calculus is often used in formal semantics to express the meaning of a sentence compositionally as a series of successive function applications. On Barbara Partee's website there's a nice lesson on how lambda calculus (or, as some say, the lambda calculus) can be used to model English semantics. Lesson 7 introduces lambda calculus, but it might be worth looking at earlier lessons to see why it is useful.
For sure, lambda calculus is not "calculus" in the typical sense of differential calculus that we learn in our high school math classes, but it is a kind of calculus in that it involves an operator (lambda) that operates on a function and returns another function, binding a variable in the process. In that sense the lambda operator gives you something like the "derivative" of the original function.
Comment: I hardly ever downvote, and have not done so here, but I think this is an almost completely unhelpful answer. Lambda calculus is not what people mean who say "calculus", and is known only in a rather specialised field. This answer is like replying to a question about dialects with a disquisition on Marxian dialectics
Comment: Probably because "calculus" by itself usually refers to differential calculus, at least as far as I know. Here, have an upvote. (Also, note the second sentence in the question.)
Comment: I'm curious as to why this answer received a downvote (it had received an upvote before that, making its score 0 as of the writing of this comment). If someone deems something in the answer inaccurate or unclear, it would be helpful to know specifically what that is so that I can improve the answer. Thanks!
Answer: There's a mathematical physicist who related graphs grammars and Feynman graphs. She comes up with the, perhaps unwarranted, conclusion that any context free graph grammar determines an insertion Lie algebra and commutative Hopf algebra.
Answer: For calculus to be useful, you first have to have some kind of continuous numeric function. This not being the case with grammar, calculus won't be illuminating for the study of unbounded dependencies or association relations: grammar is based on simple "does / doesn't" logic. However, it is in principle applicable to the study of linguistic behavior, e.g formant values of pronounced vowels, or perhaps a sociolinguistic study of the "Yeah-no" construction. In the former case I would say that the phenomenon itself (vocal tract resonances and physical movements) are fundamentally continuous. In the latter case, even though sociolinguistic changes reduce to binary do / don't choices, the relevant system of relations is so complex as to be intractable, if treated binarily, so numeric functions are the only practical way to discuss many social phenomena.
Answer: This question of using calculus or a calculus-like approach has been on my mind for some time.  It's not calculus such that you'd use on a continuous data set, such as the set of real or imaginary numbers, like you'd find in a Real Analysis course, but rather as a philosophical concept.
So let's define: Calculus: An analysis of functions and how such functions change.
Underlying Assumption: There's an underlying function that can describe the mechanism of language change in the more general sense (versus language-specific).
So for example, given a phonological system in a particular semi-isolated linguistic community, the phonemes will change or shift with respect to time (this is the hypothesis, and by "semi-isolated" I mean that contact with other languages won't be a major factor to language-change).  For a graphic on this, see the entry on "The Great Vowel Shift" in English, by Princeton: https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Great_Vowel_Shift.html
This would be the English-centric example to where a wider typological example would have to be substantiated as universal in all languages.  Supposing The Great Vowel Shift shows us the underlying mechanism or function, then if we're given the Vowel system of the English language in the 1400's, we could predict the dipthongization over the next 150 - 200 years and further shifts thereafter.
The point here is to discover some underlying phenomenon that appears linguistically universal whereby we can model linguistic change and through the model we can make a prediction as to what a language will 'look' and 'sound' like via extrapolation of such a model.
My intent with providing my example is not necessarily to answer the question, but to add to the thought process that may help lead to an answer, or better, a linguistic calculus.


Why can't cloaked vessels shoot?

Question: What prevents ships firing phasers and photon torpedoes through a cloak? I have yet to see a episode where this is brought up. It may have been mentioned in The Undiscovered Country, as I know there is a kingon ship that does it, but I have not seen that movie in years. Is it addressed there or somewhere in one of the series'?
Comment: I believe it damages the cloak, because of how it works. It also is a dead give-away as to where you are. (And since you also can't have shields with the cloak up, giving away your position is a very bad idea)
Comment: In *Star Trek: Nemesis* Shinzen's warbird was capable of firing while in cloak.
Comment: Could also be because of limited energy supply, but that's more of a writing excuse.
Answer: A cloak requires the dampening or absorbing of all internal energy emmissions while simultaneously rebroadcasting all background and directed energy emmisions "through" the ship; not a simple task.
Preparing to fire weapons, even torpedos, requires taking multiple actions that would be extremely diffcult to mask or play into the backgound sound. From charging weapon capacitors to acquiring an active target lock it is very likely that the target ship would pick up on even small clues (sudden unexplained miscellaneous energy spikes, active sensor emmissions with no apparent point of origin etc) and take appropriate precautionary measures such as raising their shields.
Comment: Good answer. While this would certainly apply in battle, where the target is already running at heightened awareness, one might expect that a ship could *potentially* fire-while-cloaked on a ship who was running under normal conditions, that is, not expecting an attack, and therefore not actively watching for the minutest EM changes.
Comment: According to Memory Alpha, typical cloaks (non-experimental) require so much power that the weapons simply cannot be fired while the cloak is active. It's not a matter of detectable EM fluctuations when you fire, it's a matter of having to turn the cloak completely off in order to fire at all.
Comment: It's also entirely possible that the cloaking field, and its dispersal/redirection of energy, would interfere energy-based weapons even if you did fire them.  This could make them less powerful/effective, or even damage the cloaked ship, depending on how the energy is redirected.  You may end up shooting a phaser straight back into your own tailpipe.
Comment: @Matt - I've rolled back your edit as it added substantial material that may not be in accordance with the original answer's intentions.  It would have been better as a comment or a separate answer.
Comment: Cool! I'll add a separate answer.
Comment: I had always assumed that to be "cloaked," you have to run with shields down to avoid giving off any energy signature. Powering up, let alone firing weapons would give one away, while shields are down.  Hadn't considered that any kind of energy-dampening devices might mess with the discharge of weapons.
Answer: Cloaking is usually done via the ship's deflector grid; the deflector shield being an energy field that is being used to envelop the entire ship.  They used them to do something that has been experimented with IRL; basically relay signals / light / etc. from one side to the other, around the cloaked ship.  A major limit, however, is this is not what the shield was meant for, and, when operating in this manner, it was not operating as a shield.  Also, the power drain was such that weapons, and even Transporters could not usually be used when cloaked. Memory Alpha 'Cloaking Device' page
There were several iterations in the development and refinement of them, over the various incarnations of the shows...  And there have been ones that could fire when cloaked.

Another advanced cloak was encountered in 2379. The Reman warbird Scimitar employed a new type of "perfect" cloaking device that did not give off any tachyon emissions or residual antiprotons, making the Scimitar completely undetectable while cloaked. It allowed the ship to fire weapons and use shields while cloaked, as well as allow the ship to drop the cloak protecting certain quadrants of the ship without de-cloaking the entire ship at once. The effectiveness of this technology was demonstrated when the Scimitar engaged the USS Enterprise-E, the IRW Valdore, and another warbird in the Bassen Rift, and was able to destroy one of the warbirds and cripple the other without suffering any significant damage. Only through repeated blind targeting, barrages of phasers and torpedoes, and the use of telepathic triangulation by Commander Deanna Troi was the Enterprise-E able to overcome the Scimitar's cloak. (Star Trek Nemesis)

Also, not even going into the movies, if you recall the final episode of TNG, 'All Good Things', you saw a Klingon ship fire while cloaked -- apparently they had resolved that issue, although it being at least a partially alternate time-line, it's canon status is questionable.
Comment: If you remember in `Undiscovered Country` the Klingon's had a war BoP that was able to fire when clocked, but had no shields when clocked. This was how the assassins where able to sneak upto the Enterprise and Klingon flag ship and frame Kirk/Enterprise for the attack Bean to the Klingon Flag and kill the Chancellor. But as said by Keith it was not perfect, and got a torpedo shoved up there tail pipe, so all and all it was not that they can't it was that it was not a smart more to be someplace that you may get shot at and not have shields. IMHO
Comment: If I remember correctly the Kligons had a BoP in Enterprise that could also fire while cloaked, but its been awhile.
Comment: @Vaughn Ya know I think this may be the best explanation. It's not that you can't do it while cloaked, its the retaliatory shot that will come back at you after they detect your position and having no shields that prevents it.
Comment: -1 for "cannoninity".   http://www.thefreedictionary.com/canonicity
Answer: Some can.
In Star Trek VI: The Undiscovered Country, the Klingons built a special Bird of Prey that can fire its weapons while cloaked.
In the Series That Shall Not be Named, the ISS Enterprise from the Mirror Universe is able to fire while cloaked ("In a Mirror Darkly")
As for what you're referring to, typical cloaking devices simply require too much power to allow many other high-power systems to operate concurrently. It's not a matter of weapons making the cloak less effective; it means the weapon system simply cannot fire while the cloaking device is drawing power.
According to Memory Alpha:

Due to the enormous amount of power required to generate a cloaking field, there is by and large not enough power available to also power the weapons and shields...However, there  have been several times when advances in cloaking technology have rendered these tactical inefficiencies untrue.
Answer: An 'out-of-universe' answer: This is due to the Law of Conservation of Ninjutsu:

In any martial arts fight, there is only a finite amount of ninjutsu available to each side in a given encounter.

You need an awful lot of Ninjutsu to both hide yourself from the enemy and to land attacks on him at the same time. And the premise of most shows is that the protagonists have some Ninjutsu, but just enough to get themselves out of a pinch at the last moment.
Note: This is not the standard mode of application of the law, as you will see when you follow the link. There might be a slightly better-fitting trope...
Answer: My understanding has always been one of limited energy supply, at least in the TOS era.  As both weapons/shields and cloak required enormous amounts of energy to use, their power systems could not maintain both at the same time as they maintain life support, which is another huge energy requirement.
There was a novel about another experimental Klingon ship.  I can't recall what the title was as I read it some 15 or 20 years ago.  They took a D7 battlecruiser and ripped out all the life support systems, replacing it with additional power generators.  By eliminating the need to support a crew, they were able to generate enough power to supply power to cloak, weapons and shields simultaneously.  The ship was linked via remote control to a Bird of Prey.  The end result was a battlecruiser that could fire while cloaked, which was controlled by the Bird of Prey crew, which remained safely cloaked and hidden a short distance away while the battle went on, without needing to reveal themselves.  So, at least in this instance, it was explained as a power issue.
Answer: In Star Trek VI, the main baddie captains a Klingon Bird of Prey that CAN fire while cloaked. It only ever fires torpedoes, either because:

Disruptors use more energy, which is needed for cloaking,
A torpedo, once fired, doesn't point back to its source like a beam or pulse weapon would, or
Torpedos simply do more damage.

Anyway, when a torpedo is fired, it illuminates the skin of that Bird of Prey momnentarily; the energy of the shining ball of destruction is simply too much to overcome.
The reason most others wouldn't be able to fire is mostly because to fire requires bringing targeting systems to bear and otherwise using a lot of energy, which doesn't mix well with trying not to show up visibly or on any sensors. However, apparently transporters DO work while cloaked; this is demonstrated in Star Trek III and in ST: Generations, as well as several episodes of TNG.
Answer: It is said that the cloak draws too much power to simultaneously use weapons.
...which is a terrible explanation seeing as we know cloaked vessels can reach high warp, which one would think bending space and time requires much more power than launching a torpedo. Torpedoes by their very nature are self-propelled and self-guided, even if they are less effective without the launching ship's assistance, one would think simply launching them would be easy.
So the real answer is, in terms of plot, the status quo is almighty and a cloaked vessel that can use weapons freely while cloaked would make a race --or even a single ship-- too powerful to overcome. Such a vessel is not seen until the Reman Warbird Scimitar in Star Trek Nemesis).
Answer: In Star Trek III search for Spock the Klingon captain beams to the ship from the planet while the ship is cloaked. Pretty neat trick since they can't use transporters while cloaked
Comment: Hi @Kidjames, welcome to this website! Even if it can be argued that beaming a klingon on a ennemy vessel leads to some battle quite fast, I am not sure that actually counts as "shooting"... Do you have other examples on-topic? :)
Answer: Some can.
In Star Trek VI: The Undiscovered Country, the Klingons built a special Bird of Prey that can fire its weapons while cloaked.
In ST:ENT, the ISS Enterprise from the Mirror Universe is able to fire while cloaked ("In a Mirror Darkly")
According to Memory Alpha:

Due to the enormous amount of power required to generate a cloaking
  field, there is by and large not enough power available to also power
  the weapons and shields...However, there have been several times when advances in cloaking > technology have rendered these tactical inefficiencies untrue.
Comment: This doesn't answer the question.
Comment: Hi Matt, I took the liberty to move your old answer here, as it was much more detailed. Feel free to roll back if you disapprove. Otherwise, please consider linking to the specific M.A. page where you took the quote.
Answer: Chris is 100% correct.  The ONLY reason that ships cannot fire while cloaked is because it would seriously damage the plot device.  Let's be perfectly honest.  Star Trek VI is set in 2293 and the Klingons (whose technology is generally inferior to the Federation's) managed to build a prototype ship even then.  It's impossible to conceive that the technology was possible in 2293 but not in 2358 when the Federation had the phase cloak and the Romulans had their D'Deridex Type-B Warbirds with their almost limitless amounts of energy from their quantum singularities. The phase cloak in particular is far more advanced than any Romulan or Klingon cloaking device before or after (since you literally CANNOT be detected or fired upon while phased). With that technology, a vessel doesn't need extra power to maintain a phase, it just needs extra power to change its phase. All that would be needed is for a ship to be phased with its cloak and have smart torpedoes that change their phase to our regular continuum milliseconds before impact.  Not only would it hit, but the target vessel would be COMPLETELY unaware that anything was amiss. If done correctly, the target ship could be destroyed with a volley and no one on the target ship would have had time to send out a distress call. In fact, battle is probably more humane this way because the target vessel wouldn't even have time to know anything was wrong until it was all over.  See the problem here?  The plot device is destroyed before the story even begins. (Although it would have been a great weapon against the Borg.)
Mind you, there is one really good thing about a working phase cloak with these kind of combat tactics.  The Galactic Empire wouldn't stand a chance against the Federation with this weapon.  Even a single Oberth-class vessel could easily overpower even an Executor-class Star-Destroyer or a Death Star.  They can phase their ship through the shields and hull, right to a weak point (like the main reactor).  Once there, they could fire one of their "phase-quantum torpedoes" (patent pending :P), sit back and watch with satisfaction as the whole place goes up with them completely safe from the explosion in their phased state. The Empire will never know what had happened.  They would have no choice but to chalk it up to technical malfunction and execute a random engineer for it.
I hope that this both informed and entertained you.  It's a slow day at work. XD

Why can't cloaked vessels shoot?


Is it possible to read and/or modify the SMS header on Android?

Question: I am developing an Android application for 1.6 and above that sends and receives SMS messages through a port.
To send the SMS I am using the method:
SmsManager.sendDataMessage(String destinationAddress, String scAddress, short destinationPort, byte[] data, PendingIntent sentIntent, PendingIntent deliveryIntent)
but have discovered that the byte[] data should contain only the message body. There seems to be no method available to send multi-part data messages and no way of modifying the SMS header (UDH) to specify that the messages is, say, 1 of 3.
I'm having a similar problem on the receiving end when trying to extract the message count information from the SMS I have received through the specified port. I am using SMSMessage.getPdu() to extract a byte[] containing the message header and message body but have no way of extracting the specific information from the header data.
Does anyone know the format the Android platform uses for SMS headers?
Answer: You will need to check 3GPP SMS spec. For me, I got the destination port info from 30th and 31st byte.


12.04 doesn't recognize my wifi Broadcom BCM94318MPG

Question: I have a Gateway MX6436 with a Broadcom BCM94318MPG WiFi. Everything else works fine but Ubuntu doesn't recognize my WiFi. It says it doesn't have the firmware.
What to do? Please help.
Answer: Install firmware-b43-installer from the Ubuntu Software Center.
Ubuntu already includes drivers for Broadcom hardware, but for certain reasons can't include firmware. This has to be done by the user, as shown above.


Greatest common divisor applications

Question: What are some real-life applications of gcd? I am looking for a motivating way of introducing this topic in an elementary number theory course.
Comment: "Simplifying a fraction" is essentially dividing numerator and denominator by their gcd, e.g., simplifying $12/18$ is done by dividing numerator and denominator by $\gcd\{12, 18\} = 6$ to get $2/3$.
Comment: @FranzLemmermeyer I am not sure what the intent of your comment is. OP actively seeks to educate themselves on the subject, in that sense it seems they agree that it is good to know this when teaching ENT.  Further, it is not uncommon that instructors learn a subject they teach 'on the fly' (either by choice or as circumstances dictate it); this can work well. I would like to invite you to express your thoughts on the subject in a more constructive and supportive way.
Comment: Am I the last person on this planet who thinks that people who don't know applications of greatest common divisors should not be teaching number theory?
Comment: I'm not sure what you mean.  I mean ... pretty much *everything* in number theory depends in one way or another on this concept.  I often tell classes that even if people hadn't discovered prime numbers, they would have had to discover relatively prime as a concept.
Comment: As mentioned in the answers below, computing the gcd is equivalent to computing the lcm, and the lcm shows up whenever you have 2 (or more) periodic occurrences; the lcm tells you how often they happen together. So everything from analyzing sinusoidal waves to determining the next time you will run out of both peanut butter and jelly involves the lcm (and hence the gcd).
Answer: Another classic is the following:

A rectangular floor measures $300 \text{ cm} \times 195 \text{ cm}$. What is the largest square tiles that can be used to cover the floor exactly?
Answer: Going back to Euclid, I have found questions such as `Given a large supply of rods of length $15$ and $21$, what lengths can be measured?' can appeal to students. This also motivates the result $\gcd(a,b) = ra+sb$ of the (extended) Euclidean Algorithm.
Comment: Can also apply to stamp or coin problems.
Comment: Also applies to liquid measurement given too containers, say of 5 and 7 gallons. Advantage: you can invoke Die Hard 3.
Answer: Some nice geometrical applications arise in the analysis of periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Concrete experience with implementations in toys like Spirograph also provides excellent motivation for more abstract concepts such as cyclic groups.
Comment: This is a good one, and it's how I introduced $\gcd$ to a class of education students. I drew a couple of $\{n/k\}$ star polygons and handed out a sheet with 12 sets of 12 vertices of a regular dodecagon for creating the various $\{12/k\}$. It was moderately successful, asking them to figure out how many "pieces" the star polygons would have, and what relationship between $n$ and $k$ would cause the star to be connected.
Answer: Here's a word problem for the greatest common divisor:

12 boys and 15 girls are to march in a parade.  The organizer wants them to march in rows, with each row having the same number of children, and with each row composed of children with the same gender.  What is the largest number of children per row that satisfies these constraints?

There should be $\operatorname{gcd}(12,15)=3$ children per row ($4$ rows of boys and $5$ rows of girls).
Also, I know that you didn't ask for it, but when I teach the gcd I also teach the least common multiple.  Here's a word problem for the lcm.

A child is having a birthday party and knows that either 12 or 15 guests would attend it.  The child's father wants to buy candies for the guests, with each guest having the same number of candies, but doesn't want to buy more candies than is necessary.  What is the minimum number of candies that the father should buy?

The father should buy $\operatorname{lcm}(12,15)=60$ candies, so that if $12$ friends come, they each get $5$ candies, and if $15$ come, they each get $4$.
Comment: I'm not sure if my "contrived" examples can be considered "applications," but they might inspire more "realistic" examples.
Answer: I think for the most elementary level of introduction Joel and Benjamin has provided good examples. I think I should try too.

$1.$ When we have a linear diophantine equation like $ax+by=c$ where $a,b \ and \ c$ are integers, we get integer solutions in $x,y$ iff $(a,b)|c$.
$2.$  As a rearrangement of a well theorem of divisibility, we know that for two numbers $a,b$ we have $[a,b]=\frac{|ab|}{(a,b)}$. So, ultimately knowing G.C.D. of two numbers give us their L.C.M and vice versa.
$3.$ This one may overflow for a high school student because the way it is designed, consider the question:
Two regular polygons are inscribed in the same circle. The first polygon have $1982$ sides and second have $2973$ sides. If polygons have any numbers of common vertices, how many such vertices there will be??
The solution is simply the common roots of the equation $z^{1982}-1=0$ and $z^{2973}-1=0$ which in turn equals gcd$(1982,2973)=991$.
Answer: The special case of $gcd(a,b) = 1$ (i.e. $a,b$ are relatively prime) is in its own right an important motivation for $gcd(a,b)$, especially if you link it with the (extended) Euclidian algorithm. Students tend to like the idea of secret codes. You can connect the notion of being relatively prime with the notion of a number $a$ being invertible mod $b$ and show how the Euclidean algorithm can efficiently compute this modular inverse, and then point out how this is the crucial step in computing private/public key pairs in RSA encryption.

Greatest common divisor applications