While I have a lot of sympathy with the view that it is not really fair to unnecessarily price something above where most people want to pay for it, what you say is not actually true.
If you're selling an object (or a bit string, or whatever) you have to set one price for it. A price for the rich and another for the poor does not fly.
It can, sadly, be the case that setting a price of X and getting ten thousand customers does not make as much profit as setting a price 2X and getting one thousand.
Take an example, you have Toyota who make cars. They make a small profit on each individual car, but they sell a lot. You also have Rolls Royce. They make a huge profit on each car. They could sell their cars much cheaper and still make a profit - and if they did a few more people would buy - but they have decided, probably correctly, that they will do better to sell at the price they are selling at.
It is similar with media products. If Warner thought they can make more money lowering prices and selling more stuff, they would. They wouldn't deliberately lose money for the hell of it.
The effect of people being able to copy stuff is that the people who would be willing to pay the price if they had no other option, do have an option and some will use it. So there is some loss (or if you prefer: less profit than without copying). Of course, there is the counterbalance that more illegal copying can lead to people finding out about new artists and buying stuff they wouldn't otherwise have bought, and of course the RIAA idea that every downloaded copy of a song is the price of a single lost is pure crap (it's a tiny fraction of that).
If a phone book publisher lists a business which they know to be a scam they can be done for accessory to wire fraud (doubt it would happen though). The dividing line is pretty simple: if you are doing something which you could reasonably believe is not facilitating crime (e.g. six months ago publishing a phone book listing Bernard Madoff's investment business) then you are not culpable.
If it is clear that you know that you are publishing something which facilitates illegal activities, particularly if you are making money from it, you are likely to get in trouble. If a drug dealer pays you to make him flyers advertising his coke, don't take the money.
You might remember that Bram Cohen had a search facility running at bittorrent.org at one point - which did list some stuff that turned out to be copyrighted; but he would remove stuff when this was pointed out (similar to YouTube). He gave up because this was too much work and removed the feature - but he was not prosecuted for it. Because he did not know there was illegal content available until he was told, and when he was told he removed the links. Even though in the time between something appearing on the search and someone asking him to remove it, quite a lot of people had downloaded stuff they weren't legally allowed to.
The Pirate Bay, by contrast, hasn't removed anything, and they have been told... They really can't claim they have been doing anything but deliberately helping people break the law, and they were making at least some money from it.
While probably this is going to end up in some stupidity like lawsuits against torrent software, TPB really doesn't have a leg to stand on. Yes, sure they aren't actually hosting copyrighted material themselves, but they do make money. And they make money by enabling people to breach copyrights.
Their defence holds up about as well as a pimp coming out with 'But I am not a prostitute! I just show the clients the girl and take some money!'
Hate to tell you this... but you are in fact doing the copyright gravy train thing.
You're just proxying it off to whichever journals you publish in, to which you transfer a (usually limited) copyright. They make quite a bit of money selling subscriptions, and unless you get journal publications you won't get a job or a grant.
That's true but sadly almost impossible to work. People routinely make preprints available as soon as they submit to a journal (a few weeks before it goes to a referee, often) and many people will give talks about their work when they're sure of the correctness of the result - which can be a very long time before it's written up. I gave a bunch of talks last summer on a couple of subjects, neither of which is yet written up; one because we think we can do something more, and the other because all of my coauthors are busy and we're just taking time to get it written down. When these papers go to a referee, the referee will probably have heard me or a coauthor talking about the stuff, and they'll recognise who wrote it whether or not our names are on the paper.
That said, I don't think there's so much discrimination against the guy you've never heard of; or at least, I would not write a bad report unless the paper was bad. What there is, is pressure not to reject a big name even if they are submitting something uninteresting (that should go to a minor journal) or very badly written (where normally I would want to list 20 problems with the first 2 pages then state that the paper must be fully rewritten before I accept it) to a major journal.
Depends on what area you're in, I think. I suppose there are some areas where really only a few people work and it's so different to anything else that no-one from outside can easily referee a paper - but if you're in that position, I'm hoping that you're working in a very new and growing subject. The alternative, bluntly, is that you're doing something the rest of the world thinks is boring.
I do combinatorics; if you give me a paper that doesn't use topological or heavy probabilistical methods, I could referee it with a bit of work; that covers about half of combinatorics (i.e. thousands of people). If you cut that back to the bits of combinatorics where I've actually published (so am likely to get referee requests in turn) then you still have a few hundred active researchers.
It's certainly true that sometimes you can guess who wrote a review - but mainly if they have a very distinctive writing style: Nash-Williams was famous for insisting on correctness in English to the point that his referee report could easily end up longer than the paper, for example. Since most referee reports are quite short - one paragraph about whether the result looks correct and is good enough for the journal, and a list of bullet points to change - it's usually hard to guess who wrote it.
Spam yes. Crackpots piped to math.GM for the amusement of all (e.g. the guy whose 'proof' of the Riemann hypothesis was 20 pages of verbiage boiling down to 'the universe is built on maths and maths is built on primes, so they must behave naturally and therefore the result is true..').
Journal publications are basically used to tell people who don't work in your area that you're doing decent work. If someone is doing decent work in an area you're a specialist in, you probably know them at least by sight and you probably hear about their results fairly soon after they prove them; the journal paper may well come a year or two later.
But if you want funding, or you want a job, you have to convince a bunch of people who know very little about your area that you are a valuable person. The easiest way to do that is to point at recent papers in good journals (which, really, isn't so different to the web of trust idea: I have a paper in CPC because someone thought my work was good enough to go there, that kind of thing).
There are lots of problems with the sort of metric you suggest; you need something relevant to now, you don't want it to discard people who do good work on their own or in tight groups (and there are quite a few of the latter), you don't want it to be distorted by the sort of mathematician who will publish every result they can get in any collaboration (there are quite a few, some of whom are very good and very well-connected but still publish some boring results along with the good ones).
Summary is certainly not accurate: the paper doesn't claim any kind of miscalculations; the published papers giving confidence limits do explicitly assume truth of models.
That said...
The paper essentially says: if the models are accurate then there is a tiny probability of catastrophe - but if the models are inaccurate then that probability might be much larger. Which is fine as far as it goes, it's certainly in accordance with standard probability theory.
However it does miss one quite important point: which is that whether or not the 'true' model of physics is what we think it is (in fact, we know we don't have it) it has to obey certain conditions. It has to be true that the intermediate-scale low energy limit is Newtonian, that if you introduce high energies, large masses or simply large scale then the limit must be Einsteinian, and that the low-energy small-scale limit must match known quantum theory. All these things are tested literally billions of times a day. The only questions are with what exactly happens in the areas we haven't tested much (which admittedly are large areas).
This means that even if there are errors in the 'accepted models' used to calculate the chance of catastrophe, those errors are very likely not to be enormous - if you prefer, the paper should perhaps split its analysis into 'accepted model holds', 'accepted model is only wrong by a factor of 10^3' and 'accepted model is badly wrong' - and the point is that the probability of catastrophe given either of the first two cases remains tiny under the existing analysis, while we may reasonably assume that the probability of the third case is miniscule.
As to the withdrawing of papers - yes, many papers have flaws, and even many flaws remain undetected. On the other hand, many papers are not really read in detail - and those papers tend to coincide. If some graduate student writes a paper on something not especially interesting, then they will read it (but it's hard to catch your own errors), their supervisor should read it (but may not do so properly) and the referee should read it (but is likely to simply check for plausibility not go into detail). Quite possibly no-one else looks beyond the abstract and first few pages - so errors aren't caught. On the other hand, if a paper is important and many people look at it, then errors usually are caught because several people independently check the details in the process of trying to understand it.
One should not push too far the 'if all our theories are wrong there will be a catastrophe' idea - it's not a false idea, but equally it's perfectly possible that when the 9 billion names of God are written down the universe will end; it's just not very likely.
You can make it fairly foolproof, but you can't make nuclear power terrorist-proof except by having people around guarding the nuclear plant. That works when you have a few big power plants serving lots of communities, but if you want lots of little plants then you will not be able to guard them all. And then one day you'll wake up to find a terrorist has put a car bomb into one of these little plants and spread radioactive material all over a city. Build new big plants, not lots of little ones.
It is still a slightly strange thing to do in maths.
I've done some decent work on my own, and some decent work in collaborations. It's a lot more fun to work with people, and you usually get more done. That said, if I had an idea about how to solve a really big conjecture, I would not start looking for people to work with, I'd try to solve it on my own first.
Physicists rank authors, 'first author' is the PhD student who did the work, second author is the lecturer who helped with the student's problems on the way, third is the official supervisor, fourth is the lab head who needs to be kept happy for more money next year.
Mathematicians do not rank authors, names in alphabetical order always.
You'd lose your bet, massively. There are really not that many proofs-by-computer in maths; and for those that do exist, the interesting part is not the computer's part.
To give an example, the Four Colour Theorem has a computer-assisted proof; but the computer's part basically consists of generating and colouring a bunch of maps according to rules Haken and Appel programmed in. But it's not all that interesting to see 1500 or so maps which can all be four-coloured, we already knew you can colour lots of maps with four colours, it's something that schoolchildren try out in primary school. What is interesting is the proof that if you are given a huge map - even perhaps one with millions of regions, far bigger than any of the maps the computer coloured - then the only ways that the huge map could not be four-colourable would be if one of the 1500 small maps wasn't. That part of the proof was not done by computer.
This is not to say I don't like computer proofs (even if I'd rather see a short elegant proof), but mostly maths problems either break down into just a few cases - and it's quicker to solve them by hand than program a computer - or they don't break down at all, or they do break down into cases but the number of cases is too big for computers (Lehel's conjecture, for example, could be proved by checking 2^{161999000} cases...). There just aren't many problems which turn out to break down into too many cases to do by hand but not so many that it's out of reach for computers.
It still depends on the speed your gears are turning. If you don't gear things up off the windmill (i.e. you try to have your sculpture working at 1 rotation of mill = 1 rotation of all gears) then you'll get quite a lot of torque - probably not 550flb, but more than will work with anything cheap bar used car parts. If on the other hand you do gear things up (which will require heavy stuff, but only near the windmill) so the windmill spins once and your gears spin 20 times, then you'll be fine with much lighter gears. Simple equation: torque times RPM is a constant on your sculpture (it's the power, albeit in some strange units). If you gear up so the RPM increases, the torque decreases and you can use lighter gears. It'll also make the movement smoother.
You keep saying this, but HP isn't really the point. Spin a cheap plastic gear at 20k RPM and you can probably get >1HP through it without too much trouble, but spin it at 10RPM with 0.05HP on it and you'll be lucky to see it last a minute before the teeth break. You need to care about the force being applied to the gear teeth - and 550flb is quite a lot of torque, so little gears will probably break, thin axles will twist out of true or simply shear. Two options - go for heavy stuff, or gear things up so you don't need the torque.
For the top end of the torque requirement, yes - or go to a junkyard and disassemble a few gearboxes and trans shafts.
For any smaller more delicate bits, where you do not need that sort of torque, try to find a box of old-style (metal) Meccano on ebay / some car boot sale, and buy some rods of the right width (1/8 inch?) to get decent length axles.
One that's missing: on a motorway, find a van doing 80 and sit behind it. Assuming there's no jams on the motorway (and you are willing to stay alert) you are not going to need to brake to avoid running into the back of the van, and you are going to find your fuel economy goes up, a lot: like 40 up to 60mpg.
Incidentally, accelerating hard burns a lot of fuel. It's only worth doing when you're on a busy road and will get no slipstream during the time when you're going much slower than everyone else.
Depends what field you work in. It certainly gets all the junk (there was someone claiming a poly-time algorithm for SAT recently...) but it also gets a lot of first-rate stuff that the author wants to get published right now, which he'll send to a journal later. If this guy (who at least seems to have got some help from some fairly big names, so presumably isn't a complete nut) is worried that maybe someone else might be working on the problem and getting results, then the natural thing to do is get it out ASAP.
Assuming it's all right (not my field...) then it will go in a journal (assuming this guy is more normal than the last guy to solve one of these big conjectures: Perelman did just drop his paper on the ArXiV, and refused to publish in a journal or accept prizes). But it will not appear for a year or two, so it needs to be put up somewhere anyway.
You're not too far off in terms of general development - but you really cannot assign ages like this.
Some kids develop faster, others slower. If you look more closely, it's usually even more of a mixture: some kids learn some things faster and other things slower. I still remember my first primary school teacher insisting that at age 5 I could not possibly have learnt to read yet, and not allowing me to have books beyond 'A is for Apple' when I wanted to have something more like 'Thomas the Tank Engine' (not so much more advanced, maybe, but there are complete sentences in the latter even if they're short). Three weeks of boredom seems like a lot when you're five (that being about how long it took her to understand that I could read simple sentences without sounding out the words).
(incidentally - sibling = (brother or sister), not child)
I didn't even mention NP-completeness. Mainly because it has nothing to do with the problem. Are you just slinging buzzwords on a subject you don't understand?
Basically, no. The proof essentially says: whatever target state you want to get to, we will regress to some natural bottom state then go to the target state.
If you prefer, it's a bit like deciding to go from London to Paris by taking the first flight out of every airport you get to, starting at Heathrow, carrying a toy gun and a French passport. Eventually you'll end up on a flight to America, where they stop you as a terrorist and (by way of Guantanamo Bay) return you to the address on your passport which is in Paris. You get there eventually, but it takes a few years, and pretty much every node switch is massively sub-optimal.
'deterministic automaton' basically means: computer program which doesn't have any form of randomness involved. If you're in this particular state in the program and you do this particular thing then always the same thing happens.
The road colouring problem amounts to: For any program, I claim that there is one single sequence of actions so that if you do this sequence of actions, from whatever start point, when you finish you will be in a specific state (`target', say) of the program.
If you take any real normal program, it's `obvious' what to do: for example if you're looking at 'edit' and you want a typed copy of Shakespeare in Times 12pt, you hit backspace as many times as edit can have characters (lots, but it's a finite number) then you go through the menus by keypress and set each style to the correct thing, then you start typing in the works of Shakespeare which eventually gives you the target state. And it doesn't matter if the monitor was off, you know that this method definitely works, whatever state edit was in when you got there, whether it was in the blank just loaded state, or whether it was editing font size in War and Peace, whatever.
It is not so easy to prove that in fact for any program you can find such a sequence.
In fact, it isn't even true. A really simple example is the program which every time you press a key changes the screen between red and blue. Now you're supposed to give a sequence of actions - keystrokes - which are guaranteed to end up with the screen blue (target state). But really what the keystrokes are is irrelevant. This program doesn't care if you press 'a' or 'ESC', it just changes screen colour every stroke. So really your sequence of actions comes down to 'press keys however many times'. How many times? Well, if the screen starts red (and remember the monitor is off, so you don't know if that's true) then you'd need to press keys an odd number of times to get it to end up blue. So you have to do that. But then what happens if the screen started blue? It ends up red. This sequence of actions doesn't always reach the target state - and there cannot be any sequence of actions which does.
So you have to impose a condition to make the result true. In the example I gave, the program has two states and it loops between them. It's a program with exactly one loop, of length two, and the greatest common divisor (GCD) of these loops is of course 2. Basically, having the ability to loop between states like this (or in a loop of length 3, or 4, or whatever) is the only 'obvious' barrier to having a sequence of actions which goes to a target state. So you impose a condition: the GCD of all the loops is 1 (this doesn't mean there are no loops, but the guy proves it means you can find a way out of all the loops). If you don't have that, then you can find automatons (programs) which work like the example and you cannot possibly have the sequence of actions you need. If you do have it, there is no 'obvious' barrier to the sequence of actions existing. And what this paper shows is that there isn't any hidden catch: once you get rid of the 'obvious' barrier then you can find a sequence of actions which definitely puts the program you're looking at into the target state.
Unfortunately, it really doesn't matter whether the code has commercial value. You work for Company X, they have a bit in the contract saying whatever you produce belongs to them, and they can enforce it if they want to, whether it's valuable or not. This has been around for a long time, there are lots of precedents. In fact, typically even if you didn't produce the code on company time you'd be stuffed. If you wrote it in your lunch breaks on a company computer (and let's assume you somehow have proof you were on lunch breaks) you don't legally have clear ownership: you'd be required to cut the company in on any profits, and you'd be liable if you open-sourced it.
Of course, like you say you might get permission to open-source it from a boss, and you might (if it's not big, and you're going to do a load more work anyway so it's not very recognisable) just open source it. Though the latter is risking trouble later.
What most studies show is that if you take a person who is overweight and normally doesn't exercise (which is a big fraction of the population) and you tell them to 'do some exercise' then they will not lose any significant amount of weight.
It doesn't say if they will end up still fat but with a bit more fitness (which is at least good for not having heart attacks running for buses).
It also says nothing about what happens if you take a person who is willing to work hard enough that it's not comfortable (most people won't push themselves) for a decent length of time. I usually eat about 3-4000 calories a day (depends if I exercise, if I don't I'm not so hungry). In fact I've just finished eating a big chocolate pudding. On the other hand, I'm about to go cycling for an hour or two, and by the time I've done that, I'll have burnt off all those calories and my weight is not likely to go up from its current 80kg. In fact, since it's winter and cold I have to eat more food than I really want in order to avoid losing weight.
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While I have a lot of sympathy with the view that it is not really fair to unnecessarily price something above where most people want to pay for it, what you say is not actually true.
If you're selling an object (or a bit string, or whatever) you have to set one price for it. A price for the rich and another for the poor does not fly.
It can, sadly, be the case that setting a price of X and getting ten thousand customers does not make as much profit as setting a price 2X and getting one thousand.
Take an example, you have Toyota who make cars. They make a small profit on each individual car, but they sell a lot. You also have Rolls Royce. They make a huge profit on each car. They could sell their cars much cheaper and still make a profit - and if they did a few more people would buy - but they have decided, probably correctly, that they will do better to sell at the price they are selling at.
It is similar with media products. If Warner thought they can make more money lowering prices and selling more stuff, they would. They wouldn't deliberately lose money for the hell of it.
The effect of people being able to copy stuff is that the people who would be willing to pay the price if they had no other option, do have an option and some will use it. So there is some loss (or if you prefer: less profit than without copying). Of course, there is the counterbalance that more illegal copying can lead to people finding out about new artists and buying stuff they wouldn't otherwise have bought, and of course the RIAA idea that every downloaded copy of a song is the price of a single lost is pure crap (it's a tiny fraction of that).
If a phone book publisher lists a business which they know to be a scam they can be done for accessory to wire fraud (doubt it would happen though). The dividing line is pretty simple: if you are doing something which you could reasonably believe is not facilitating crime (e.g. six months ago publishing a phone book listing Bernard Madoff's investment business) then you are not culpable.
If it is clear that you know that you are publishing something which facilitates illegal activities, particularly if you are making money from it, you are likely to get in trouble. If a drug dealer pays you to make him flyers advertising his coke, don't take the money.
You might remember that Bram Cohen had a search facility running at bittorrent.org at one point - which did list some stuff that turned out to be copyrighted; but he would remove stuff when this was pointed out (similar to YouTube). He gave up because this was too much work and removed the feature - but he was not prosecuted for it. Because he did not know there was illegal content available until he was told, and when he was told he removed the links. Even though in the time between something appearing on the search and someone asking him to remove it, quite a lot of people had downloaded stuff they weren't legally allowed to.
The Pirate Bay, by contrast, hasn't removed anything, and they have been told... They really can't claim they have been doing anything but deliberately helping people break the law, and they were making at least some money from it.
While probably this is going to end up in some stupidity like lawsuits against torrent software, TPB really doesn't have a leg to stand on. Yes, sure they aren't actually hosting copyrighted material themselves, but they do make money. And they make money by enabling people to breach copyrights.
Their defence holds up about as well as a pimp coming out with 'But I am not a prostitute! I just show the clients the girl and take some money!'
Hate to tell you this... but you are in fact doing the copyright gravy train thing.
You're just proxying it off to whichever journals you publish in, to which you transfer a (usually limited) copyright. They make quite a bit of money selling subscriptions, and unless you get journal publications you won't get a job or a grant.
That's true but sadly almost impossible to work. People routinely make preprints available as soon as they submit to a journal (a few weeks before it goes to a referee, often) and many people will give talks about their work when they're sure of the correctness of the result - which can be a very long time before it's written up. I gave a bunch of talks last summer on a couple of subjects, neither of which is yet written up; one because we think we can do something more, and the other because all of my coauthors are busy and we're just taking time to get it written down. When these papers go to a referee, the referee will probably have heard me or a coauthor talking about the stuff, and they'll recognise who wrote it whether or not our names are on the paper.
That said, I don't think there's so much discrimination against the guy you've never heard of; or at least, I would not write a bad report unless the paper was bad. What there is, is pressure not to reject a big name even if they are submitting something uninteresting (that should go to a minor journal) or very badly written (where normally I would want to list 20 problems with the first 2 pages then state that the paper must be fully rewritten before I accept it) to a major journal.
Depends on what area you're in, I think. I suppose there are some areas where really only a few people work and it's so different to anything else that no-one from outside can easily referee a paper - but if you're in that position, I'm hoping that you're working in a very new and growing subject. The alternative, bluntly, is that you're doing something the rest of the world thinks is boring.
I do combinatorics; if you give me a paper that doesn't use topological or heavy probabilistical methods, I could referee it with a bit of work; that covers about half of combinatorics (i.e. thousands of people). If you cut that back to the bits of combinatorics where I've actually published (so am likely to get referee requests in turn) then you still have a few hundred active researchers.
It's certainly true that sometimes you can guess who wrote a review - but mainly if they have a very distinctive writing style: Nash-Williams was famous for insisting on correctness in English to the point that his referee report could easily end up longer than the paper, for example. Since most referee reports are quite short - one paragraph about whether the result looks correct and is good enough for the journal, and a list of bullet points to change - it's usually hard to guess who wrote it.
Spam yes. Crackpots piped to math.GM for the amusement of all (e.g. the guy whose 'proof' of the Riemann hypothesis was 20 pages of verbiage boiling down to 'the universe is built on maths and maths is built on primes, so they must behave naturally and therefore the result is true..').
Journal publications are basically used to tell people who don't work in your area that you're doing decent work. If someone is doing decent work in an area you're a specialist in, you probably know them at least by sight and you probably hear about their results fairly soon after they prove them; the journal paper may well come a year or two later.
But if you want funding, or you want a job, you have to convince a bunch of people who know very little about your area that you are a valuable person. The easiest way to do that is to point at recent papers in good journals (which, really, isn't so different to the web of trust idea: I have a paper in CPC because someone thought my work was good enough to go there, that kind of thing).
There are lots of problems with the sort of metric you suggest; you need something relevant to now, you don't want it to discard people who do good work on their own or in tight groups (and there are quite a few of the latter), you don't want it to be distorted by the sort of mathematician who will publish every result they can get in any collaboration (there are quite a few, some of whom are very good and very well-connected but still publish some boring results along with the good ones).
Summary is certainly not accurate: the paper doesn't claim any kind of miscalculations; the published papers giving confidence limits do explicitly assume truth of models.
That said...
The paper essentially says: if the models are accurate then there is a tiny probability of catastrophe - but if the models are inaccurate then that probability might be much larger. Which is fine as far as it goes, it's certainly in accordance with standard probability theory.
However it does miss one quite important point: which is that whether or not the 'true' model of physics is what we think it is (in fact, we know we don't have it) it has to obey certain conditions. It has to be true that the intermediate-scale low energy limit is Newtonian, that if you introduce high energies, large masses or simply large scale then the limit must be Einsteinian, and that the low-energy small-scale limit must match known quantum theory. All these things are tested literally billions of times a day. The only questions are with what exactly happens in the areas we haven't tested much (which admittedly are large areas).
This means that even if there are errors in the 'accepted models' used to calculate the chance of catastrophe, those errors are very likely not to be enormous - if you prefer, the paper should perhaps split its analysis into 'accepted model holds', 'accepted model is only wrong by a factor of 10^3' and 'accepted model is badly wrong' - and the point is that the probability of catastrophe given either of the first two cases remains tiny under the existing analysis, while we may reasonably assume that the probability of the third case is miniscule.
As to the withdrawing of papers - yes, many papers have flaws, and even many flaws remain undetected. On the other hand, many papers are not really read in detail - and those papers tend to coincide. If some graduate student writes a paper on something not especially interesting, then they will read it (but it's hard to catch your own errors), their supervisor should read it (but may not do so properly) and the referee should read it (but is likely to simply check for plausibility not go into detail). Quite possibly no-one else looks beyond the abstract and first few pages - so errors aren't caught. On the other hand, if a paper is important and many people look at it, then errors usually are caught because several people independently check the details in the process of trying to understand it.
One should not push too far the 'if all our theories are wrong there will be a catastrophe' idea - it's not a false idea, but equally it's perfectly possible that when the 9 billion names of God are written down the universe will end; it's just not very likely.
You can make it fairly foolproof, but you can't make nuclear power terrorist-proof except by having people around guarding the nuclear plant. That works when you have a few big power plants serving lots of communities, but if you want lots of little plants then you will not be able to guard them all. And then one day you'll wake up to find a terrorist has put a car bomb into one of these little plants and spread radioactive material all over a city. Build new big plants, not lots of little ones.
Don't worry, you're not senile, everyone forgets to zip down sometimes...
It is still a slightly strange thing to do in maths.
I've done some decent work on my own, and some decent work in collaborations. It's a lot more fun to work with people, and you usually get more done. That said, if I had an idea about how to solve a really big conjecture, I would not start looking for people to work with, I'd try to solve it on my own first.
Physicists rank authors, 'first author' is the PhD student who did the work, second author is the lecturer who helped with the student's problems on the way, third is the official supervisor, fourth is the lab head who needs to be kept happy for more money next year.
Mathematicians do not rank authors, names in alphabetical order always.
You'd lose your bet, massively. There are really not that many proofs-by-computer in maths; and for those that do exist, the interesting part is not the computer's part.
To give an example, the Four Colour Theorem has a computer-assisted proof; but the computer's part basically consists of generating and colouring a bunch of maps according to rules Haken and Appel programmed in. But it's not all that interesting to see 1500 or so maps which can all be four-coloured, we already knew you can colour lots of maps with four colours, it's something that schoolchildren try out in primary school. What is interesting is the proof that if you are given a huge map - even perhaps one with millions of regions, far bigger than any of the maps the computer coloured - then the only ways that the huge map could not be four-colourable would be if one of the 1500 small maps wasn't. That part of the proof was not done by computer.
This is not to say I don't like computer proofs (even if I'd rather see a short elegant proof), but mostly maths problems either break down into just a few cases - and it's quicker to solve them by hand than program a computer - or they don't break down at all, or they do break down into cases but the number of cases is too big for computers (Lehel's conjecture, for example, could be proved by checking 2^{161999000} cases...). There just aren't many problems which turn out to break down into too many cases to do by hand but not so many that it's out of reach for computers.
It still depends on the speed your gears are turning. If you don't gear things up off the windmill (i.e. you try to have your sculpture working at 1 rotation of mill = 1 rotation of all gears) then you'll get quite a lot of torque - probably not 550flb, but more than will work with anything cheap bar used car parts. If on the other hand you do gear things up (which will require heavy stuff, but only near the windmill) so the windmill spins once and your gears spin 20 times, then you'll be fine with much lighter gears. Simple equation: torque times RPM is a constant on your sculpture (it's the power, albeit in some strange units). If you gear up so the RPM increases, the torque decreases and you can use lighter gears. It'll also make the movement smoother.
You keep saying this, but HP isn't really the point. Spin a cheap plastic gear at 20k RPM and you can probably get >1HP through it without too much trouble, but spin it at 10RPM with 0.05HP on it and you'll be lucky to see it last a minute before the teeth break. You need to care about the force being applied to the gear teeth - and 550flb is quite a lot of torque, so little gears will probably break, thin axles will twist out of true or simply shear. Two options - go for heavy stuff, or gear things up so you don't need the torque.
For the top end of the torque requirement, yes - or go to a junkyard and disassemble a few gearboxes and trans shafts.
For any smaller more delicate bits, where you do not need that sort of torque, try to find a box of old-style (metal) Meccano on ebay / some car boot sale, and buy some rods of the right width (1/8 inch?) to get decent length axles.
One that's missing: on a motorway, find a van doing 80 and sit behind it. Assuming there's no jams on the motorway (and you are willing to stay alert) you are not going to need to brake to avoid running into the back of the van, and you are going to find your fuel economy goes up, a lot: like 40 up to 60mpg.
Incidentally, accelerating hard burns a lot of fuel. It's only worth doing when you're on a busy road and will get no slipstream during the time when you're going much slower than everyone else.
Depends what field you work in. It certainly gets all the junk (there was someone claiming a poly-time algorithm for SAT recently...) but it also gets a lot of first-rate stuff that the author wants to get published right now, which he'll send to a journal later. If this guy (who at least seems to have got some help from some fairly big names, so presumably isn't a complete nut) is worried that maybe someone else might be working on the problem and getting results, then the natural thing to do is get it out ASAP.
Assuming it's all right (not my field...) then it will go in a journal (assuming this guy is more normal than the last guy to solve one of these big conjectures: Perelman did just drop his paper on the ArXiV, and refused to publish in a journal or accept prizes). But it will not appear for a year or two, so it needs to be put up somewhere anyway.
You're not too far off in terms of general development - but you really cannot assign ages like this.
Some kids develop faster, others slower. If you look more closely, it's usually even more of a mixture: some kids learn some things faster and other things slower. I still remember my first primary school teacher insisting that at age 5 I could not possibly have learnt to read yet, and not allowing me to have books beyond 'A is for Apple' when I wanted to have something more like 'Thomas the Tank Engine' (not so much more advanced, maybe, but there are complete sentences in the latter even if they're short). Three weeks of boredom seems like a lot when you're five (that being about how long it took her to understand that I could read simple sentences without sounding out the words).
(incidentally - sibling = (brother or sister), not child)
I didn't even mention NP-completeness. Mainly because it has nothing to do with the problem. Are you just slinging buzzwords on a subject you don't understand?
Basically, no. The proof essentially says: whatever target state you want to get to, we will regress to some natural bottom state then go to the target state.
If you prefer, it's a bit like deciding to go from London to Paris by taking the first flight out of every airport you get to, starting at Heathrow, carrying a toy gun and a French passport. Eventually you'll end up on a flight to America, where they stop you as a terrorist and (by way of Guantanamo Bay) return you to the address on your passport which is in Paris. You get there eventually, but it takes a few years, and pretty much every node switch is massively sub-optimal.
The road colouring problem is not 'an equation'.
'deterministic automaton' basically means: computer program which doesn't have any form of randomness involved. If you're in this particular state in the program and you do this particular thing then always the same thing happens.
The road colouring problem amounts to: For any program, I claim that there is one single sequence of actions so that if you do this sequence of actions, from whatever start point, when you finish you will be in a specific state (`target', say) of the program.
If you take any real normal program, it's `obvious' what to do: for example if you're looking at 'edit' and you want a typed copy of Shakespeare in Times 12pt, you hit backspace as many times as edit can have characters (lots, but it's a finite number) then you go through the menus by keypress and set each style to the correct thing, then you start typing in the works of Shakespeare which eventually gives you the target state. And it doesn't matter if the monitor was off, you know that this method definitely works, whatever state edit was in when you got there, whether it was in the blank just loaded state, or whether it was editing font size in War and Peace, whatever.
It is not so easy to prove that in fact for any program you can find such a sequence.
In fact, it isn't even true. A really simple example is the program which every time you press a key changes the screen between red and blue. Now you're supposed to give a sequence of actions - keystrokes - which are guaranteed to end up with the screen blue (target state). But really what the keystrokes are is irrelevant. This program doesn't care if you press 'a' or 'ESC', it just changes screen colour every stroke. So really your sequence of actions comes down to 'press keys however many times'. How many times? Well, if the screen starts red (and remember the monitor is off, so you don't know if that's true) then you'd need to press keys an odd number of times to get it to end up blue. So you have to do that. But then what happens if the screen started blue? It ends up red. This sequence of actions doesn't always reach the target state - and there cannot be any sequence of actions which does.
So you have to impose a condition to make the result true. In the example I gave, the program has two states and it loops between them. It's a program with exactly one loop, of length two, and the greatest common divisor (GCD) of these loops is of course 2. Basically, having the ability to loop between states like this (or in a loop of length 3, or 4, or whatever) is the only 'obvious' barrier to having a sequence of actions which goes to a target state. So you impose a condition: the GCD of all the loops is 1 (this doesn't mean there are no loops, but the guy proves it means you can find a way out of all the loops). If you don't have that, then you can find automatons (programs) which work like the example and you cannot possibly have the sequence of actions you need. If you do have it, there is no 'obvious' barrier to the sequence of actions existing. And what this paper shows is that there isn't any hidden catch: once you get rid of the 'obvious' barrier then you can find a sequence of actions which definitely puts the program you're looking at into the target state.
Unfortunately, it really doesn't matter whether the code has commercial value. You work for Company X, they have a bit in the contract saying whatever you produce belongs to them, and they can enforce it if they want to, whether it's valuable or not. This has been around for a long time, there are lots of precedents. In fact, typically even if you didn't produce the code on company time you'd be stuffed. If you wrote it in your lunch breaks on a company computer (and let's assume you somehow have proof you were on lunch breaks) you don't legally have clear ownership: you'd be required to cut the company in on any profits, and you'd be liable if you open-sourced it.
Of course, like you say you might get permission to open-source it from a boss, and you might (if it's not big, and you're going to do a load more work anyway so it's not very recognisable) just open source it. Though the latter is risking trouble later.
What most studies show is that if you take a person who is overweight and normally doesn't exercise (which is a big fraction of the population) and you tell them to 'do some exercise' then they will not lose any significant amount of weight.
It doesn't say if they will end up still fat but with a bit more fitness (which is at least good for not having heart attacks running for buses).
It also says nothing about what happens if you take a person who is willing to work hard enough that it's not comfortable (most people won't push themselves) for a decent length of time. I usually eat about 3-4000 calories a day (depends if I exercise, if I don't I'm not so hungry). In fact I've just finished eating a big chocolate pudding. On the other hand, I'm about to go cycling for an hour or two, and by the time I've done that, I'll have burnt off all those calories and my weight is not likely to go up from its current 80kg. In fact, since it's winter and cold I have to eat more food than I really want in order to avoid losing weight.