applications of Tate-Poitou duality

Question: What are nice applications of Tate-Poitou duality?
Answer: One of the main themes of current number theory research, instigated by the work of Wiles and Taylor--Wiles on the Shimura--Taniyama modularity conjecture for elliptic curves, is the study of modularity (or more generally automorphy) of representations of Galois groups.   The basic tool for proving modularity is the so-called Taylor--Wiles method, which involves introducing certain well-chosen auxiliary primes to aid in analyzing a generalized Selmer group (a certain
Galois cohomology group) which controls the infinitesimal structure of the space of Galois
representations.
In applying this method, one relies on a certain formula (due to Greenberg and Wiles) which
expresses the ratio of the orders of a Selmer group and the corresponding dual Selmer group
in terms of a product of local terms, and the proof of this formula is an application of the
Poitou--Tate exact sequence.  See Washington's article in "Modular forms and Fermat's
Last Theorem" for an exposition of the Greenberg--Wiles formula.
There are lots of similar applications of Poitou--Tate in Iwasawa theory.  Whenever one studies Selmer groups, one confronts the problem of trying to analyze a global object
which is defined in terms of local conditions.  The Poitou--Tate exact sequence provides a relationship between local and global Galois cohomology groups, and so is fairly ubiquitous in the study of Selmer groups.
Answer: Take a look at the recent paper of Mazur and Rubin, "Ranks of twists of elliptic curves and Hilbert's 10th problem"
Lemma 3.2, one of the indispensable lemmas of the paper, is a direct application of the Poitou-Tate exact sequence.
Comment: +1, that's actually an excellent example, both in terms of the importance of the result and the importance of the role Poitou-Tate plays there.
Answer: There are loads of applications and what you consider "nice" hugely depends. One application
is e.g. Tate's proof that the Birch and Swinnerton-Dyer conjecture is invariant under isogenies. That and more applications are contained in Milne's book on "Arithmetic Duality Theorems".
Answer: Three examples of the use of the Poitou-Tate duality:
All parity results (Dokchitsers, Mazur-Rubin, Nekovar, ...) for elliptic curve use this duality somewhere.
The duality is also crucial for Euler systems. An Euler system produces a collection of "derived" global cohomology classes. Using the pairing induces by Poitou-Tate, one can show that these classes are not in the Selmer group but orthogonal to it. This then bounds the size of the Selmer group from above.
The Poitou-Tate duality is an analogue of the Poincaré duality. For instance, Tate reduced the conjecture of Birch and Swinnerton-Dyer over a global field of characteristic $p$ to the finiteness of Sha. This argument uses Poincaré duality on the corresponding ellitic surface over the finite field. Analogous in Iwasawa theory, one uses the (Cassels-)Poitou-Tate duality in the computation of the leading term of the characteristic series of the Selmer group of the elliptic curve.
I would suspect there are more examples in "Cohomology of Number Fields".
Answer: I think that Galois cohomology of elliptic curves by Coates and Sujatha is a good place to see some of the applications to Iwasawa theory as mentioned by prof Emerton.
Answer: It has applications to the Brauer-Manin obstruction for torsors under connected linear algebraic groups, see Sansuc, J.-J., Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80 for number fields and Diego Izquierdo, Dualité et principe local-global sur des corps locaux de dimension 2, http://www.math.ens.fr/~izquierdo/Localdim204052016.pdf

applications of Tate-Poitou duality


Dumping workflow persisted data?

Question: On a large SharePoint Farm, we have several worflows designed.
Some of them are Visual Studio Workflow.
When something goes wrong in the workflow only few evidence are visible (some in the uls logs, and sometimes in the workflow history).
I would like to be able to detail what data is persisted in the workflow, in order to diagnose what's wrong.
Is there any way to dump the workflow persisted state?
I've tried using Powershell :
<code>$web = Get-SPWeb http://my_sp_web
$list = $web.Lists["MyList"]
$item = $list.Items.GetItemById(42)
$wf = $item.Workflows[0]
</code>
But unfortunately, my <code>$wf</code> variable does contains anything useful.
Answer: I'm not sure you can do that.
Why not add a better error handling?
http://msdn.microsoft.com/en-us/magazine/dd419656.aspx
Best regards,
Comment: You are right. This would be the better solution. Unfortunately it's hardly applicable immediately. There is a very large code base, and each modification should be budgetised, developed, tested, etc. This won't solve my immediate issue.

Dumping workflow persisted data?


Does inclusion of significant quotations from literature make a novel harder to publish?

Question: I'm considering using quotations (a sentence to a paragraph) from various pieces of literature  at the beginning of each chapter of my novel. The quotations connect thematically to the events of the chapter. However, I'm concerned that this might make a publisher less likely to take my work-- either because it's atypical or because there might be copyright issues. Is a publisher likely to consider dealing with the need to get permission for quotes? If I only use works in the public domain, are there still potential issues? Thanks!
Answer: Check your contract (if you already have one). In some cases, responsibility for securing permissions rests with the author. If so, the publisher will not be concerned so long as you deliver the evidence that you have all the approvals.
If you believe that the extracts from other works are essential to your novel (and it was once a fairly common practice) but your publisher is wary of extra work, you could offer to take on the role of obtaining all the relevant releases to remove that potential "barrier".
Answer: If you're just using short quotes, like a sentence or two, I think this would clearly fall into "fair use" and not require any permission. If you're quoting more than a paragraph, and your purpose is just to connect your work to some bigger theme, you're probably quoting too much and should cut it down. I take it that you are not trying to create an anthology of classic literature, but to write your own book and connect it to classic literature. If that's the case, then you shouldn't have any problem. I suppose if your quotes are long, a publisher might worry that you are "on the edge" between fair use and a copyright infringement. At that point you could either cut down the length of the quote or get permission. Getting permission might cost money.


Sharepoint2007: Display Document Library Folder contents in customized format?

Question: How can I do a customized view of a Document Library in home page?
I want to show like following:
<code>![image][1] **Title of Document** [with link to document]
            Body of articles [Intro Text]
</code>
Best Regards,
Reds
Answer: When you customize a view, SharePoint allows you to choose among different styles. In your case, you could try a "Summary View", or a "Boxed View".
If none of the out of the box styles suits your needs, then you'll have to use SharePoint Designer for advanced layout customizations. In SharePoint Designer you'll work with a Data View Web Part (SharePoint 2007) or a List View Web Part (SharePoint 2010).
Comment: I want your answer to up vote but my reputation can't :)
Answer: There are several ways to do this, as with all things SharePoint. My favorite is with a Data View Web Part (DVWP), which you can create in SharePoint Designer. Because DVWPs are XSL-driven, you have total control over the markup, structure, and look of the output.


How to create a one to one correspondence between two sets?

Question: I am stuck with,

Give a one to one correspondence between Z+ and positive even
  integers.

Now, I don't have an idea how to show that there is a one to one correspondence between the two. I would be thankful for some hints.
Comment: Please avoid using images in that fashion. How hard is it to type that in?
Answer: When asked to prove that "$\exists$ a ..."; then what you are really doing is actually finding whatever is that you need to prove exists.
For example in your question you must prove that there exists a one-to-one correspondance between $\mathbb{Z}^{+}$ and positive even integers, which I will now denote $\mathbb{Z}_e$ so we should attempt to find a map which takes $\mathbb{Z}^{+} \rightarrow \mathbb{Z}_e$.
So how should we go about finding one? well first lets think what is the formal definition of even? I would say an integer $x$ is even if $x = 2k$ for some $k\in \mathbb{Z}$ so the set of positive even integers is $\mathbb{Z}_e = \{x = 2k : k\in\mathbb{Z}^{+}\}$.
Now once we have actually formalized what a positive even integer is it is not hard to think of a map, for example take:
$f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}_e$ defined by :
$k \mapsto 2k$
Now we've got a map we think we will work, and we just need to check if it is one-to-one.
Suppose $f(r) = f(s)$ Then $2r = 2s$, but this quickly implies that $r = s$ so the map is one-to-one, as desired.
Furthermore the map is also onto, because $\mathbb{Z}_e = \{x = 2k : k\in \mathbb{Z}^{+}\}$ is the set of integers of the form $2k$ by definition.
Answer: Consider the f(x)=2x. Now if f(a)=f(b) then 2a=2b then a=b ( because $2\neq0$) this proves the function is injective, to prove that is surjective a positive even integer is of the form 2k with k a positive integer, then f(k)=2k.


Mobile SharePoint site navigate only to the top level site

Question: We are having a problem with our mobile sharepoint site. The links to the subsite takes us back to the top level site when clicked on mobile devices such iPads, iPhones, Blackberry smart phones, and on all other smart phones.
For example:
We have a top level site named MyCompany and three subsites (HR, SALES, and Marketing). Whenever we click on the subsite the link takes us back to the home page of MyCompany.
We used firefox, safari, Opera, & IE but still getting the same issue. This doesn't happen when we access it on notebooks and PCs though.
PS: It only happens when we access the link outside of the company network.
Any ideas?
Thank you.
Answer: I got it to work. Cross-firewall access zone wasn't set up. I just created alternated access mapping and mapped that zone to cross-firewall access zone.
Thank you.


Hurricanes and bathtubs

Question: Why do hurricane winds rotate? I know it´s about Coriolis force but how can I be quantitative? My problem is that I can´t  imagine the direction velocity of wind and the angle with earth angular velocity rotation. Also, when can we explain other things like bathtubs, toilets.. using Coriolis? I mean when this force is relevant to explain the motion and the direction of rotation, CCW or CW(clock wise)?
Comment: The bathtub part is a possible duplicate of http://physics.stackexchange.com/q/7738/2451 and http://physics.stackexchange.com/q/32/2451
Comment: Not the Hurricane analysis
Answer: Sure coriolis force applies, but I think a much simpler intuitive explanation is conservation of angular momentum.
Think of the spinning skater who pulls in his/her arms & legs, spinning faster.
If you look down on the earth from the north star, you see a whole hemisphere covered with air rotating counterclockwise.
At the pole, it's basically not moving, and at the equator it's travelling to the East at about 1000 miles per hour,
so that air has huge angular momentum in the counterclockwise direction.
Anything that pulls some of it together toward a point is going to increase its angular speed relative to the air around it, and vice versa.
In a hurricane you have a strong updraft of warm air, which pulls surface air toward its center, causing it to rotate counterclockwise.
It gets into the inner vortex and rises, coming out at the top.
As it goes outward at the top, it slows down, so appears to rotate in the opposite direction with respect to the overall air mass.
So you see clouds going counterclockwise at the bottom, and clockwise at the top.
Southern hemisphere, it's just the opposite.
In a bathtub, the rotation of the earth is much less likely to be effective, compared to the way the water just happens to have been stirred.
P.S. Angular momentum is mvr, where m is mass, v is tangential velocity (meters/second), and r is radius from the center. Since it is conserved, if you reduce the radius by a factor of 10, you increase the tangential velocity by a factor of 10 (and the angular velocity (radians/second) by a factor of 100).

Hurricanes and bathtubs


Why does a pacifier trigger my baby's gag reflex?

Question: My baby doesn't often take a pacifier, but occasionally he will. I've noticed lately that once in a while, putting in the pacifier will trigger his gag reflex and will sometimes even cause him to spit up. I've tried using the "mam" pacifier that we received in the hospital, as well as some Avent ones... he's been using all of these happily until now, although not often. The gagging is a new thing and only occurs sometimes... What could be causing it? Is there anything to do about it?
Comment: Is there any reason to keep giving him a pacifier if he isn't enjoying it?
Comment: @HedgeMage +1 for that - I wish we'd toughed it out a bit more in the beginning with baby #1, because in the end it was a tough habit to get him to kick. With #s 2 & 3 on the way, we're doing all we can to avoid pacifiers altogether. If your child doesn't seem to enjoy a pacifier, perfect - skip them.
Comment: @HedgeMage - that's why we don't give it to him often. But occasionally he does seem to want it and enjoy it... except lately this gagging thing is getting in the way. Also, he's started to suck his thumb lately, and I'd prefer addiction to a pacifier over addiction to a thumb, considering that you can't take a thumb away :-/
Answer: We had a similar problem, especially with Avent, and you can just try brands until it works. It may start happening now simply because you baby is sucking more strongly, or the pacifier is getting old and stretchy.
Our daughter seemed to prefer pacifiers that was really wide, but ultimately she rejected the idea of pacifiers altogether. :-)
Answer: I can think of a few possibilities.  One is that some babies prefer some brands; mine did not like anything but Nuk brand pacifiers.  It surprised me; I know kids learn brands cause they're bombarded by advertising, but I didn't think it started THAT young.  Another possibility is that your baby just doesn't like/want pacis.  I don't know if I'd push it in that case, unless perhaps you're flying and want the baby to suck on something to help pop his ears.
The other possibility I can think of is that there is some kind of tongue-tie or palate issue.  From my experience on blogs and message boards for women who could not breastfeed, I have discovered that problems with the baby's mouth structure are surprisingly under-diagnosed.  Here is some info on tongue tie and high palate, although finding info on high palate outside breastfeeding blogs seems to be a bit difficult.  I'd bring it up with the baby's pediatrician on the off chance that the baby has some sort of issue with his mouth.
Answer: My oldest son never liked them, he would just spit them out.  My youngest loves them, and whenever he can he looks for them, although when he was younger if the pacifier was a little too long it triggered something similar to what you describe.  Lennart is right, you just need to try different brands until your child finds one that fits or wait until your child grows a little and the pacifier will probably fit better.
Answer: All pacifiers are not created equal! The design of the nipple either encourages, interferes with or compensates for lack of development of normal oral motor control.
The gag is a protective reflex and is usually triggered by touch to the posterior 1/3 of the tongue. If the pacifier is too long it will naturally elicit a gag in children with normal reflexes. Perhaps your child's sucking reflex is developmentally becoming stronger and therefore he's more sensitive. Enlarged tonsils can cause a gag reflex to be more sensitive also. And some children are less tolerant of oral stimulation.
A broad and flat nipple can slow development of tongue grooving which is an important for controlling a liquid or food bolus for swallowing.
My son gagged on all nipples and meat consistency baby foods. He is 28 and a college graduate and is still a very picky eater that cannot tolerate more textured foods. This is one of the differences that he was born with that makes him unique.
Since you mention that he sometimes spits up, I wonder if reflux episodes could be the reason he is gagging. Perhaps he wants the pacifier because he's experiencing some indigestion and sucking is calming his system, until the reflux surfaces causing the gag and spit up.
Trying shorter or different pacifiers, monitoring for signs/symptoms of reflux, and rule out enlarged tonsils are all reasonable approaches to address this behavior.
Answer: You should ask your pediatrician about gagging.  The great news is that while they are a handy invention, babies were fine before pacifiers existed.  Suckling instincts can be satisfied by suckling, and they're likely suck a thumb for a while regardless of pacifier use, so that is something you'll have to deal with eventually either way.  Listening to the issues of families that used pacifiers (including the children "requiring" it before they are able to return it to the mouth themselves, and later weaning), I'm glad I didn't bother.
Comment: You're not answering the question.
Comment: Suggesting professional advice and providing an alternative that will prevent gagging (not using a pacifier) are each reasonable answers to the question.


How can I view only one task list in Jorte?

Question: Suppose I'm at the store, and all I want to view is my "Stuff to Buy at the Store" list in Jorte. How can I view only that list instead of all the tasks on any list?
Answer: In the application, click on the 'Tasks & Memos' area. There should be a Task list button; click on it.
There, you can select and deselect the lists you want to show. If you only wanted to show a single list, you can uncheck the rest of the lists and click Close.
Now, this isn't designed to be used to quickly show and hide lists. Rather, it seems more like a preference. If you want multiple lists and the ability to quickly load just one on demand, I would recommend an app designed specifically for lists, such as List Master.
I use Jorte, but only for my calendars. It isn't well suited for separate lists, but rather a simple todo list.


How do you put out a grease fire?

Question: I recently had a grease fire - the pan caught on fire from a little fat that spilled over the skillet. Is there a safe, quick way to put out a fire like this?
Comment: If this happens frequently, e.g. more than once in your life time, then I say it worth investing some money on a fire extinguisher.
Comment: @user3528438 In any situation where you aren't trying to trade your home for insurance money, you should have an extinguisher (edit: or a pile of accessible salt) ahead of time as cheap insurance against burning everything down and harming loved ones and innocents. It doesn't matter how often you think the event might occur: accidents are by their nature accidental and you'll need to replace the extinguisher after a single use anyway. It isn't like it will ever pay off in any way except stopping that first fire.
Answer: Steps to put out a grease fire

Turn off the source of heat. If the burner / element controls are behind the fire, turn off the gas main or cut electricity at the circuit breaker. Turn off any fan blowing or pulling air around the fire.
Do NOT pour water on it.
Let me say that again: DO NOT pour water on it!
Attempt to remove all oxygen from the flame. You can cover it with its lid, with another pot or wok, or with a baking pan, anything nonflammable. Fire blankets, however, usually cannot handle grease fires.
If you can't cover it, dump baking soda on it... but you need lots of it. Some people below had success with salt. Heated baking soda releases carbon dioxide, starving the fire of air if it's cool enough that it won't reignite. Salt just absorbs the heat and smothers it like a lid. Do not use flour or any flammable powders. Do not throw the baking soda or salt in a way that might spread the fire.
Secret step number 6: If all else fails, use a fire extinguisher. Class K extinguishers are low pressure and meant for grease fires. Aim any other kind of high-pressure extinguisher indirectly so that it won't spread the fire. If the fire seems contained by step 4 or 5, get the extinguisher ready but don't use it if the fire is dying out on its own.
Comment: @Tree77 : a word of warning; I was cooking with my neighbor when he had a greese fire in the burner, and we tried covering it with a lid over the burner;  It **did not** work, because on his stove, there was enough of an air gap from under the burner (the whole top of the stove lifted up so you could clean underneath) that the fire kept going, and burned up his stove.  If we had gone straight for the fire extinguisher, rather than smothering, we might've been able to save the stove.
Comment: I only keep a single box of baking soda in the kitchen - probably not enough for firefighting.  Don't use flour instead of baking soda.  Flour can be quite flammable.
Comment: For home cooking, this is the right answer.
For anything bigger, you want a halon extinguisher (preferably an automatic one) or some Aqueous Film Forming Foam (AFFF).
It's amazing what you learn in the navy, isn't it?
Comment: Surely "fire blanket" would be an option before extingusher...
Comment: @Carmi : You can't get halon in all areas.  As a backup, you want something that's rated well for 'class B' fires, but I'd also recommend bouncing it off of the wall rather than spraying straight at the fire, to disipate the energy and not push the fire around, making it larger.
Comment: +1 for attempt to remove oxygen, 2 days ago this happened to me, the fire quickly went out on its own but if it had not, I had no baking soda and no fire extinguisher. Removing oxygen would have been my only option.
Comment: This answer misses: in a modern kitchen with an air extraction hood (it that  the correct English word?), switch that off. You don't want that excessive heat (or flames) drawn into your air channels.
Comment: @Joe: Obviously, you're right, and not everything is correct everywhere. The main point is that if the fire is hot enough to be burning grease, you can't dissipate the heat fast enough, so you need to smother the fire with soemthing.
The advantage of halon is that once the fire is out and everything cools off, you can just vacuum the stuff away and it doesn't do any damage.
There is a whole art to putting out fires by spraying net to them to dissipate heat and avoid causing thermal shcok to things, but that takes serious training.
Comment: @KatieK haha I tried it with flour. Fortunately it didnt light and after the grease fire was out I had a big pile of smoldering flour and had to cover it with a wet towel (which I should have done in the first place).
Comment: @RowlandShaw Fire blankets may work, if you know how to apply them, but a [2014 test from a Belgian consumer organization](https://www.consumentenbond.nl/fornuis/blusdekens-in-de-fik-bij-frituurbranden) showed that 13 out 16 tested fire blankets sold commercially are not heavy duty enough for grease fires. They actually [recommended NOT using those](http://frituurbrand.nl/). (Sorry, all Dutch language links)
Answer: I'm an arson investigator and I give classes to businesses with our fire extinguisher demo. The participants must supply their own fire extinguishers but they get to practice using them in a controlled environment. Here's the rules for kitchen/grease fires that I teach. When you have a grease fire the first step is to cover the pan. If the controls are on the front of the stove you should turn off the burner, but if the controls are on the back of the stove you don't want to risk a burn reaching for it until you've covered it. Next step...nothing. Leave it alone and let it cool up for quite a while. NEVER EVER EVER move the pan. Even after you've covered it. I can't tell you how many structure fires I've investigated because someone tried to move a burning pan of grease off of the burner. It's almost impossible to move a pan full of any liquid without sloshing or spilling it, then add the excitement of the fire and you're almost guaranteed to end up spreading grease everywhere. Grease and oil heat up to over 300 degrees F. so it will take quite some time to cool off but don't risk moving it. Keep it covered and it will cool off on it's own.
There have been several people mentioning all the "home remedies" to extinguishing grease fires. Never use the wet towel method. Water boils at 212 degrees F. and cooking oils are commonly used at temperatures well above that. The oil will quickly boil off the water in the towel and then you will be left with an oil soaked wick. Halon has been mostly restricted to a few specific fixed extinguishing systems and I haven't seen them used as portable fire extinguishers in a long time. Class B rated fire extinguishers probably won't be very effective due to the higher temperatures which is why Class K fire extinguishers have become the standard agent specifically for extinguishing grease fires. The extinguishing agent in Class K fire extinguishers can withstand the high temperatures and forms a soapy film on top of the grease, it acts like a lid restricting oxygen from the fuel. Class B fire extinguishers will work briefly but the high heat breaks down the agent before the grease can cool below it's ignition point and once the oxygen is allowed to mix with the grease vapors it will re-ignite. Dry Chemical fire extinguishers (not to be confused with dry powder which is for flammable metals) can work on grease fires, but as with most fire extinguishers the agent is expelled at high pressures that will usually splash the grease and spread the fire. Class K fire extinguishers expel the agent at much lower pressures which avoids the splashing effect. Water should never be used in a grease fire. Water boils at 212 degrees F. When it turns into a vapor it expands 1700 times its volume as liquid water. Since grease is well above this temperature when it is burning the water instantly boils and expands causing the grease to be thrown out of the container spreading the fire. Google some frozen turkey - turkey fryer videos to see examples of this.
Baking soda will work, if you can avoid splashing, which is why I don't recommend it. Dry Chemical fire extinguishers, commonly known as ABC fire extinguishers, usually have some form of sodium bicarbonate as the dry chemical agent. Never use flour, it's very flammable. I'm sure there are tons of YouTube videos of this for you to see. The problem with most powders are that when you get any substance in a small enough particle, lots of substances become flammable or explosive. 'Ladder 49' was based off of a grain silo fire which is a very real scenario. It was grain dust that was generated from pulverizing grain in the conveyor process of moving the grain around. Sugar is also explosive in dust form. Google the Georgia Imperial Sugar refinery explosion of 2008.
TL,DR:

Cover the fire with a lid or another pan to exclude oxygen from the grease vapors
Turn off the burner or heat source
Leave it alone until it has cooled off, Do not try to move it.
Class K fire extinguishers were designed specifically for grease fires and are the only fire extinguisher recommended for use on grease fires.
Don't use household powders, most are flammable/explosive when the particles are separated like they are when they are thrown. Google 'flour fireball'
Comment: I've never heard of a class K: is that perhaps the same as class F (I'm in Europe)?
Answer: I had a grease fire years ago while using my first electric stove. I dumped most of a box of baking soda on it with no visible effect; my wife ran in and hit it with a Halon fire extinguisher, which actually spread the grease and fire against the wall behind the stove! I called 911, and the dispatcher told me to dump salt on the fire. I did - about a cup - and it killed the fire almost instantly.
Baking soda puts out fires by releasing carbon dioxide and smothering the fire, but if your fire is burning violently enough, the convection created by the fire pulls in enough oxygen to offset the smothering effect. Smothering with baking soda could work if you had a bucket of baking soda, but a little refrigerator-size box won't cut it, as I learned through experience. Salt, on the other hand, apparently is a great absorber of heat for its volume. It sucks so much heat out of the grease that it simply can't support combustion anymore.
Bottom line:

Never, ever use a pressurized fire extinguisher on a grease fire.
Baking soda is probably not going to do the trick.
Salt is your friend.

It's been twenty-some years, and I've never had a kitchen fire since (knock on wood!), but I still make sure to have a box of salt within reach whenever I fry anything.
Comment: A fire extinguisher isn't necessarily bad, but you have to use it appropriately for the kind of fire you have. Standing right in front of a pan full of grease and shooting it with pressurized _anything_ will obviously spread it all over the place. You need to stand back and use the extinguisher to starve the fire. It may take a little longer than you'd expect.
Comment: But salt is 1) incredibly cheap 2) safe 3) almost magically fast.  Obviously if you haven't turned off the heat, the oil could flash again - but at least in my (one) experience, half a cup of salt killed the fire instantly.
Answer: Daniel is right. Suffocate the fire. DO NOT DO NOT DO NOT attempt to pour water on it. The water will superheat and create a huge fireball of vaporized grease and steam. People have been seriously injured.
Comment: Major +1 for the video.
Answer: I like the Mythbusters version of why you shouldn't do this better. (First 20 seconds of the video, for scary firey plume montage)
When you add water, it flashes instantly to steam and blows molten oil high into the air, where, suddenly in an O2 rich environment, it bursts into flame. Impressive, and something you never want to happen in your kitchen.
Buy yourself a high quality fire extinguisher, and learn how to turn off the electricity/gas at the source. You can try to smother it, but honestly I don't recommend it if it's anything to really worry about. Oil in a pan isn't going to burn that high. Oil accidentally knocked on the floor is going to burn your whole house.
Turn off the heat source, and wait until the flames subside. Do not move the pan. Do not put anything on or near it that may burn. Do not do anything that might splash oil out of the pan. When the oil is hot enough to burn, you don't want it out of the pan under ANY circumstances...It will IMMEDIATELY burst into flame.
Answer: Smother with lid, fire blanket or wet (wet you plonker!) cloth. Then, yes, use the extinguisher.
On the extinguisher front:
Yes, you can use an extinguisher to put out grease fires, but be careful. There are different kinds of extinguishers. Check that the one in your kitchen (or the one you're likely to use) is the right kind before any kind of fire.
Water extinguisher: Fortunately there aren't too many of these around now. Water extinguishers should NEVER be used in a kitchen. Between grease fires and electrical fires, it's almost universally the wrong extinguisher to use.
CO2 - (High pressure carbon dioxide) Extinguisher: These are the most common you'll find, and recognizable by the big "horn" dispenser. CO2 extinguishers work by removing oxygen, but can disperse quite quickly. They are also under pressure, so don't get too close to the source of fire when using a CO2 extinguisher. You don't want to "blow" the oil out of the pan. The big plus with this type of extinguisher is that the CO2 evaporates (it makes a kind of CO2 snow) so very little damage is done other than damage caused by the fire. Caveat: These extinguishers can be useless outdoors, or in highly ventilated environments. CO2 is great for really small indoor fires before they get out control. The "snow" coming out the extinguisher is cold (think dry ice) and some stuff (especially electronics) doesn't like the extreme cold, but on balance it's the best indoor extinguisher, though like all pressurized extinguishers, dangerous if improperly sprayed directly at burning liquid.
Dry-Powder Extinguisher (The powder is usually bicarb): These extinguishers works by removing the oxygen and smothering the fire. They are good for outdoors, and work on pretty much any kind of (small) fire, including kitchen fires and electrical fires. However, the retardant doesn't evaporate, and powder extinguishers do make a mess. A 9 kg extinguisher (typical size) contains 9 kg of bicarb. While these extinguishers have a "trigger nozzle", they are notoriously leaky, so  once started most of the 9 kg is going to be released. Also dry-powder extinguishers need regular (annual) servicing to prevent the contents from caking and rendering the extinguisher useless.
Obscure extinguishers: Halon and BCF extinguishers were popular because they have all the benefits of CO2, but aren't "cold," so they're a lot kinder to electronics. They're also CFCs, which have gone out of fashion for environmental reasons. And both are brutally expensive compared to CO2. Unless saving the electronics is critical, like say on a nuclear submarine, they're overkill for a home or office environment.
Bottom line: In a home situation I would recommend CO2 for most rooms, plus Dry-Powder for the kitchen.
Most importantly - if you use an extinguisher, remove the heat source as well. In the case of CO2 if the fuel is still hot, then it can reignite when the CO2 dissipates.
Lastly - after any extinguisher has been used - even for just a squirt - it must be replaced. Water and Dry Powder are "one-use" extinguishers, even with a trigger, and while control on a CO2 extinguisher is much better it must, must, must be replaced (recharged) after even a single use.
Comment: See [MikeJ's comment](http://cooking.stackexchange.com/a/35767/67) for a rebuttal to the wet towel suggestion ... it only works for really small fires, and if it's dripping wet, can make things worse.   (and this reminds me ... I need to recharge or replace my extinguishers)
Comment: +1 for mentioning the wet cloth/tea towel trick. Fast, easy, safe and effective.
Answer: Do not use water.
You have to suffocate it.  Turn off the burner (obviously),  Find a tight fitting pan lid and put it over the flame tightly.  If it's on the stove and not in a pan, you can still use a pan lid to suffocate it, you just might have to get a little creative about how you cover it.  A pan, wok or any sturdy, non-flammable cover would work as well.  Anything that can cut off their air supply.
Fire extinguishers will work as well if it's too big to fit under a pan lid, pan or other handy cover.  If you don't have a fire extinguisher and it's that big... call 911 (or your location's equivalent), you're in trouble.
Answer:

Smother the fire with a fire blanket (or any sheet of fire retardant material)
Turn off power supply / gas
Leave the blanket in place and call help if necessary
Comment: See my comment under the top answer. For *grease* fires fire blankets are not recommended.
Comment: and to follow up on @user34961 the reason is—regardless of what your poster's art shows—almost none of them are actually strong enough to handle a pot full of grease fire on their own. Maybe they'd help if all you have is one plus a heavy-duty wok: use the upside-down wok to do most of the job and the blanket to help cut off air along the sides but not trying to handle the fire itself.
Answer: when i had a fire like this i turned off the heat then pour salt over it. it sounds crazy but it worked.
Comment: Sorry you got downvoted for sharing good info when other bad advice about wet towels and fire blankets got upvoted. Generally you need to provide more context and, ideally, reliable sources like on Wikipedia.
Answer: First of all try to stay calm so you can make sound decisions. I had a pan of bacon grease spill all over the bottom of my oven so i turned it off and figured it would burn off....that is until it started to get too big for comfort. Meanwhile, i quickly drenched and totally saturated several kitchen rags and towels, and carefully opened oven. With oven mittens i quickly removed the racks and without the racks in the way, took each corner of the kitchen towel and tried to lay it evenly over the bottom of the oven. Did that with two staurated towels and the fire went out instantly. I then removed towels and after my heart rate went back to normal, i turned the oven back on andmonitored that the fire would not come back. All was good. No fire came back.
Comment: This is a different kind of fire, so you are not answering the question. AND (!), as MikeJ writes in [his answer](https://cooking.stackexchange.com/a/35767/34961) you should NEVER USE the wet towel method on a grease fire in a pan. I therefore suggest this answer gets removed ASAP.
Comment: @user34961 It really isn't a different kind of fire. Bacon grease is just lard, which absolutely gets used in the same ways as oil and shortening. Hell, it's the reason grease fires are called grease fires instead of oil or fat fires. Otoh, yeah, David should've provided more context and gone out of his way to make it clear that this only works on small fires that can quickly be completely cut off from oxygen before they burn through the cloth... and several other techniques are better if at all possible since you're risking making things much worse.

How do you put out a grease fire?


Facebook advertising: targeting for exact users

Question: Is it possible to target Facebook advertising for exact users?
For example: to show advertisements to all friends of a particular person?
Answer: It's theoretically possible to narrow down the group to about 10-20 people if you know enough about your target's profile. However, targeting a particular user's friends is the kind of privacy violation that nobody wants. The only allowed kind of "friend targeting" is showing ads to friends of your current fans, e.g. "Your friend Jack likes this page".


How to cite multiple chapters from same book in Vancouver style?

Question: How can I cite multiple chapters from the same book? Do I have to use separate reference numbers, if they are by different authors?
Answer: Since the Uniform Requirements for Manuscripts Submitted to Biomedical Journals (commonly called the Vancouver system) is also widely referred to as an Author-Number system, it will necessary to have a number for each author of a multi-chapter compilation (and more than one number for an author who produced two or more chapters).
Logically the treatment of a book to which several authors have made distinctly identified contributions should be the same as that of articles collected in a journal.
Answer: If I understand your question correctly, you want to cite from one book which is written by several authors.
Even if you know which paragraph was written by which author and you cite different authors it is still one book. The idea of listing the sources is, that a reader can order these books and look up the source.
Now, if you have six different reference numbers, because you cite six different authors of the same book, the reader will be very puzzled why he has ordered the same book six times.
Ok, probably he will figure out that theses references refer to the same book, but there is no need to puzzle him. It's one book, so use one reference number.
When you want to show that you cite different authors, you can do that in your text. Like "Mr A. Smith has written blablabla" or putting the name in parentheses after the citation or whatever. But you do not need to do that. Referencing the source in the usual way is enough.
Comment: I'm unfamiliar with Vancouver style; are the system's reference numbers really non-obvious enough that this is a concern?


Analytical calculation of the resulting surface between two overlapping spherical caps

Question: Let's say I have a sphere (determined by its center and radius) and two planes which cut individually the sphere. Individually, there will be to spherical caps. Let's suppose that both spherical caps overlap partially (i.e. part of the surface of one of the caps corresponds to a part of the other cap). There will be a body with two planar faces and a spherical face, which corresponds to the sphere. How can one calculate analytically the resulting surface area that corresponds to the spherical face of this overlapped cap?
Comment: The following question seems to be the same and has some good answers: http://math.stackexchange.com/questions/45788/calculate-the-area-on-a-sphere-of-the-intersection-of-two-spherical-caps
Answer: I would compute it as a different of total area $A = 4 \pi r^2$ and the areas of two caps.
Let $d$ denote the maximal distance from a point on the cap to the plane. Then the area of a cap equals $A_\text{cap} = 2 \pi r d$. Therefore the result would be $2 \pi r ( 2 r - d_1 - d_2)$, assuming caps planes do not intersect within the sphere.
Note also that $d = r - \text{distance from sphere's origin to the plane}$. Denote that distance by letter $h$. Then the formula for the area between two spherical caps becomes $A_\text{btw} = 2 \pi r (h_1 + h_2)$

Added: Let a plane be specified by there equations $a_1 x + b_1 y + c_1 z + d_1 =0$, and $a_2 x + b_2 y + c_2 z + d_2 =0$. The distances to these planes from the center of the sphere (assuming it sits at the origin) are $h_1 = \left| \frac{d_1}{\sqrt{a_1^2+b_1^2+c_1^2}} \right|$ and $h_2 = \left| \frac{d_2}{\sqrt{a_2^2+b_2^2+c_2^2}} \right|$ respectively.
The following numerical calculations confirm the formula and provide an illustration:
Comment: this is excellent! the only thing, in your calculations, both caps do not overlap with each other, which is my main problem
Comment: I have just updated the question to make it more clear
Answer: A warning to begin with: The morphology of the situation can vary; so it might be difficult to automatize the computing of such areas. In the following I shall deal with the simplest case: The two caps $C_i$ are less than a hemisphere, and the two midpoints $M_i$ do not belong to the intersection $S:=C_1\cap C_2$. The bounding circles $\gamma_i$ intersect in two points $P$, $Q$ (otherwise $S$ would be empty) which together with $M_1$, $M_2$ form a spherical quadrangle which is symmetric with respect to the diagonal $M_1\vee M_2$. The other diagonal $d:=P\vee Q$ divides $S$ into two pieces $S_i$ which can be treated separately. The part $S_1$ bounded by $d$ and an arc of $\gamma_1$ can be viewed as a  sector of $C_1$ minus an isosceles spherical triangle. Since all relevant angles and sidelengths can be computed from the given data, the area of $S_1$ can be found by elementary means, and the same holds for $S_2$.
This means you can forget about the highbrow procedure I proposed yesterday.
$\bigl[$The two planes $H_i$ intersect in a line $\ell$ which hits the sphere in two points $P$ and $Q$. Your piece of surface $S$ is bounded by two circular arcs $\gamma_i$ which connect $P$ and $Q$. From the given data you can compute the radii $r_i$, the constant geodesic curvatures $\kappa_i$ and the length $s_i$ of these arcs as well as the inner angle $\theta$ they enclose at $P$ and at $Q$. Given all this you can compute the area of $S$ by means of the Gauss-Bonnet theorem. You will have to take care of the signs: It makes a difference whether the shape $S$ is "convex" or is a lunula.$\bigr]$


submit the second part of a paper

Question: I couldn't find similar question being asked here. The closest one I can find is When to split/merge papers?. Here is my situation: I proved a theorem. When I try to type it, I found that it's very long. Since it's long, I splitted it into two parts. I finished the first part (50+ pages) and submitted to a journal few months ago. Now I almost finished typing the second part which is also 50+ pages. My question is:  Should I submit the second part to the same journal? If I submit to the same journal, probably the editor will send the second part to the same referee. Then the referee who is familar with the first part can read the second part more easily. However, as we all know, it's hard to publish long paper, I think it would be even harder to publish two long papers in the same journal. However, if I submit to another journal, it may be even harder since the new referee may find it difficult to read the second part without reading the first part. So this also gives me the second question: should I wait until the first part is published/accepted, and then submit the second part?
Comment: I think this question is better asked at http://academia.stackexchange.com/
Comment: I vote for same journal.
Answer: Since the two papers together prove one main theorem (if I correctly understand the first few lines of the question), it seems reasonable to submit them to the same journal.  I can imagine a referee or editor being unhappy about being asked to publish part 1, which builds up to a big theorem that will appear in a different journal.
It's true that it can be hard to find room in a journal for very long papers, but I assume the journal doesn't mind publishing papers of 50+ pages at least occasionally (otherwise you wouldn't have sent part 1 there), and the difficulty of finding space is no worse for two of your papers than for one of yours and one of someone else's.  The main effect of space shortage in such situations is that the two parts of your paper might appear in different issues of the journal.
I've published multi-part papers with all parts in the same journal. The splitting into parts was forced by the journal's upper bound on the length of papers, but I don't recall any complaints from the editors about my submitting several parts, each close to the upper bound.  In at least one such case, the parts ended up appearing consecutively in the same issue of the journal.  (The only time I've submitted "parts" to different journals was when they were really separate papers, though on the same topic.  Two parts were really logic and appeared in the Notre Dame Journal of Formal Logic, and one part was, at least in my opinion, of broader interest and appeared in the AMS Transactions.)
Comment: As a side remark, if space constraints are indeed the major consideration in this kind of decision, one would hope that moving to electronic publishing would alleviate the problem, so that Paul's decision would be primarily influenced by scholarly, rather than economic, considerations.
Answer: One option to consider, which we did with the small index subfactor papers, is to submit to the same editor but at different journals. This retains the logistical advantage of having the same referees, but doesn't require as much space in one journal. Furthermore you can put the better half in a better journal.
Answer: It is probably too much to ask from one editor to find space for two 50 page papers on the same topic, so it is natural to send the second part elsewhere. The exceptions I know involve solutions to big problems and/or big names.
About your second problem... include paper one as a helpful reference for the referee.
Answer: What I would do: Put them both on arXiv.org and then submit the second one to the same journal.
Readers will thank you (for the arXiv versions and for only having to go to one journal to read the whole result).
Comment: Ok, this is a good, kevin. put on Arxiv.org. and then submit on the similar journal.
Answer: I usually do not split in such cases, but if you do, I guess the same journal is a better choice, especially if you think of the poor people who will have to look up and reference your paper in the future :). Also, if possible at all, I would include into the first part a forward reference to the second one (if the submission is to the same journal and the second acceptance occurs before you get the galley proofs of the first part, that may be completely feasible from the technical point of view). What I really do not understand in this story is how the referee is supposed to do his job being presented with the first part alone. I can imagine myself in his shoes writing something like "The paper has many interesting ideas and no flaws so far but it is completely unclear how the ends will finally meet..."
Answer: First of all, I wonder how you can be sure that a 100-pages-proof is correct before you even have typed it in completely -- at least unless it consists mostly of routine computations!
As to your question, I think one proof should ideally be given in one paper, or a least in two papers in the same journal. -- Well, of course there are some proofs distributed over MANY papers, like the one of the Classification of Finite Simple Groups, but that's probably something different ... .

submit the second part of a paper


What kind of geometric operations "scale up" cohomology?

Question: There's an obvious operation on the category of graded rings, given by "scaling up," multiplying the grading of every element by some fixed constant.
Does anyone know of an operation on the level of spaces that will do this to the cohomology rings?
Answer: There is no such operation.  Here is one reason why.
Take your space to be CP2.  Its cohomology ring is a truncated polynomial algebra ℤ[x]/(x3), where x is an element in degree 2.
Suppose we had such a shift functor S on spaces that multiplies cohomology degrees out by n.  Then S(CP2) would have mod-2 cohomology ring a truncated polynomial algebra ℤ/2[y]/(y^3) on a generator y in degree 2n.
This violates the Hopf invariant 1 conjecture, proved by Adams.  There are no spaces whose cohomology is such a truncated polynomial ring ℤ/2[y]/(y^3) on a generator in degree k unless k is 1 (real projective 2-space), 2 (complex projective 2-space), 4 (quaternionic projective 2-space), or 8 (something associated to the octonions that I feel uncomfortable describing as projective space).
If n is not 2 or 4, you immediately get a contradiction here.  If n is 2 or 4, you could apply your shift operator multiple times and get a contradiction.
Answer: How about we change the question to this: for what kind of spaces X are there a family of  continuous maps ψk: X->X which act on cohomology by "scaling up"?  Here is a famous and bewildering example of something sort of like this:
As you can read in the notes of Sullivan's 1970 MIT course (see esp. chapter 5), if you have an algebraic variety X defined over Q(the rationals), then you get an action of G(=absolute Galois group of Q) on a space Xet=the etale homotopy type of X.
If X is a Grassmanian variety, then
* Xet has the homotopy type of the usual complex Grassmannian (up to a "profinite completion"),
* G acts on the profinite cohomology of Xet through its abelianization Gab=Z*(=the units of the profinite integers),
* this action of Gab on cohomology is by scaling.
I don't know what the motivation of the original question is, so I don't know if this kind of thing has any relation for what you want.  (I really posted this to advertise the Sullivan notes, which everybody should read.)
Answer: There isn't one.  The scaling operation that you describe (which would have to be an odd scale factor, by the way) doesn't commute with suspension.  Cohomology is a "suspension-preserving" functor (there's probably some fancy way of saying this using triangulated categories) so when you consider a cohomology theory you should really think of the target as "graded algebras over the coefficient ring with a suspension operation" only that's not quite as snappy as "graded rings" so we tend to be a bit sloppy about saying it.
Edit: This answer was a bit of a "throwaway" answer since the question felt like idle speculation (I'm a bit embarrassed that it got so many votes!).  With a little more thought I'd've concentrated on the fact that the target category isn't graded rings, or even graded algebras, or even graded algebras with suspension, but there's the action of the cohomology operations to take into account.  Simply regrading everything will completely mess up those actions.  Suspensions will come into play, though, when one considers maps between spaces and I suspect that one could show that all such maps were null-homologous.
I like Tyler's answer best.  I'm voting for that one, and I recommend it be accepted as the answer.
Comment: There is a geometric operation on spaces which sends the cohomology ring R to R ⊗ R, if we work over a field at least, and that operation on rings doesn't commute with suspension either...
Comment: Good point.  Answer edited slightly.
Answer: Even if you don't ask for the ring structure to be preserved, this seems quite unlikely, at least if you require the functor to do the same "scaling up" operation to maps on cohomology induced by maps of spaces.  I believe all the Goodwillie derivatives of such a functor would be zero, whereas "geometric" operations typically have some nontrivial radius of analyticity.
Edit: For the argument I was thinking of, I need to assume that π1(F(any sufficiently highly connected space)) = 0, so that for instance F(•) = •.
Answer: To get from spaces to graded rings you go via dg-rings.  That is,
Spaces → dg-Rings → graded Rings.
where the first arrow is the functor taking simplicial cochains, and the second is the functor taking cohomology.  So any "scaling up" of spaces should exist in dg-Rings.  But how do you scale up a cochain complex?  You can't because because that would break the degree of the codifferential.
So I don't think that there is a sensible "scaling up" for spaces.
Nice idea though, I don't know how you could classify functors that act on graded rings that can be lifted to spaces.  An obvious example might be that of setting all groups of degree greater than i say to 0.  That would correspond to filling in things in the space to kill the cohomology groups (I don't know the correct terminology off the top of my head).
Comment: Having reread Andrew's answer I think that they're essentially the same argument.  Perhaps his is the more fundamental.