matt grime

what is P^2? obvioulsy it is the space of polynomials of degree at most 2 but ot would be helpful to say so. clearly P^2 is 3 dimensional (the natrual basis being 1,t,t^2), and you have shown you have 3 linearly independent vectors: a general result and in fact the definition of dimension tells us that the set p_1, p_2, and p_3 must be a basis: any linearly independent set iwth n elements in a n n dimensional space must be a basis.

i was answering a different question, apologies, misread the request.

you certainly can ask what is the probabilty that a situation occurs in a given game such that 1/3 of the time outcome X occurs *from here*. to estimate it you need to decide at what points you're going to check the bookies' odds. Say every interval, or at the start of the day. Then look back over the history of the game and see at how many times, at the start of the interval or day, a team had a 1/3 chance of winning (presumably according to the bookies here). at least that's one interpretation of it.

iff you SUBTRACT the x^5+2x^2+x^2, why do you end up with a +2x^3 on the next line. in your forst post you wrote pseudo-ascii-math that means to divide the left hand poly by the right hand one not the other way round ie 3/5 does not mean divide 5 by 3. sorry for the confusion, it looks like a fraction. try [math] x^3+2x+1 \ | \overline{x^5+3x+2}[/math] as a makeshift solution

polynomial division is exactly as hard as dvision of integers if explained properly i think you made a typo and it ought to be [math]\frac{x^5 + 3x + 2}{x^3 + 2x+1}[/math] you are simplyfying. the easiest way is, perhaps, this. we have an x^3+2x+1 on bottom and x^5 on top. how many times does the denominator go into x^5 and what is the remainder? clearly x^5 = x^2(x^3) now if i add and subtrac cleverly then x^5 = x^2(x^3+2x+1) - 2x^3 - x^2 this is exactly the same as saying, well how many times does 14 go into 57? 57=4*14+1 so the answer is 4 times remainder 1 or 57/14= 4+1/14. back to polys. given how we can rewrite x^5, we see that the numerator is exactly x^2(x^3+2x+1) - 2x^3 - x^2+3x+2 so the whole thing simplfies to: [math]\frac{x^5 + 3x + 2}{x^3 + 2x+1}=x^2 + \frac{- 2x^3 - x^2+3x+2}{x^3 + 2x+1}[/math] now you can repeat for the higest power of x remaining

Have you considered usign latex, sarah. It will help in the long run for all the typesetting you're doing and it is quite straightforward to become proficient to such a level to write the things you need to write.

the determinant is the linear action of the inherited action of the map on the last component of the exterior algebra. does that help?

That certainly shows that any integer occuring in the sequence occurs at most once, but doesn't show that every integer is in the sequence.

so we have (w-3 > 0) => (implies) (w^2+9>6w) well X=>Y is the same as {not(X)} OR {Y} so its negation is?

do'nt use "should" as that implies that you haven't actually done the work and think it probably is true. this is maths, yo don't do that.

You'll find the hardest thing is to figure out how to "do" questions mathematically, ie with rigour. It won't matter what the topic is. For instance, what is your answer to shwoing sinh(2x)=2sinh(x)cosh(x)? Mine would read: We know that sinh(y):=(e^y - e^{-y})/2 and cosh(y:)=(e^y + e^{-y})/2, and thus it follows that 2sinh(x)cosh(x)=2(e^x-e^{-x})(e^x+e^{-x})/4 = (e^2x-e^{-2x})/2=sinh(2x) I imagine you wouldn't write as many words, or any at all. But it'll be a good habit to start writing in words what you know, what the defintions are and what you want to show, then it often pops out how to do it. And if not anyone who comes to mark it will at least know what the problem is and where you need help.

becuase that's the easiest point to expand about or we'd have horrrible expressions flyingaround with sqrt(3/4) say if we did it about 1/2. the derivative is always going to have a 1-b^2 to some fractional power and we'd have to set b to what ever the points is we're expanding about. try it. expand about 1/2 and see what you get.

yes that kind of logic is incorrect, though here it is largely a matter of presentation and gettign things in the right order. induction assumes the truth of the statements for 1,..,k and uses these to show that k+1 is true. you just showed that certain things satisfied certain relationships. it may be rewritable to correct this but the first thing you did was to declare that a_{n+1} was what we wanted to show it was. to correct it, assume it for a_{n-1} and a_{n} and then use the relation ot work out what a+{n+1 is, and tehn say this "is as required", ie don't state what a_{n+1} is before hand. it remains to check, in this case the first two cases a_1 and a_2 since we are using the "previous two" to deduce the next.

I would say the second year combines difficult work with lack of freedom of choice. Though surely hyperbolic trig is taught in high school? OF course it may be I a confusing my further maths with single maths A level. In any case hyperbolic trig and identities will not be the most difficult material you meet by a very long way, but you will when you meet the more demanding material be mathematically more mature and it won't seem as hard to learn.

well, the reason so few people understand it is that few people know all the relevant background to it. it is a huge undertaking requiring very diverse knowledge. that isn't to say that someone can't read it and check it but there is a difference between knowing something and understanding it. i know many parts of mathematics, and have read the papers proving them and found no errors in them, but I do not understand them at all. Lots of maths is "if... then...." formalism, so it is possible to understand the proof (ie to check the steps leading from the if to the then) without actually understanding the whole picture sometimes. for example, I 'know' that wiles showed via galois representations that all semistable modular forms are elliptic, and whilst i may understand the words i don't know how they fit together and I am essentially repeating a slogan. very few people understand any new result in maths.

no, the problem is that the amount of material is so huge in the case of CFSG (or in the original proof of the 4 colour theorem buried in computer code) that any errors in it are difficult to spot. imagine proof reading an article. If it's 1 page long you can get all the typos. If it is several thousand pages long then how confident are you that you got everything when you read it (and there are no computer spell checks)? Did you absolutely check every single other result that you used? There are other things too, and few articles are error free. Most errors are insiinficant, typos even, but in the CFSG case how sure are we that 15,000 pages contain no errors? it isn't that we presume it is correct per se. Any holes in CFSG are not important given the nature of the proof: break things up into smaller chunks, so errors are only on small scales). i think we can now safely say that FLT is ok. perhaps you are aware of the Clay millenium prize problems. the criteria for a successful solution is to be publsihed in a prestigious journal and to remain unchallenged for 18 months.

the thing with the CFSG (class. fin simp grp) is that no one person knows all of it; it was a collaborative effort of very many people. so far no holes have been found. some people are sure it's true and some people are less confident. i have no knowledge of any of it and have no opinion.

I hope I don't regret this since it could be taken and used as ammunition by the cranks, but for results like this, results that are huge, and cannot be easily digested by one person, then we tend to think of them as being "probably" correct. THere are known results that turned out to have no correct published proof (I figured out the correct proof for one once, but i wasn't the first to find a correct proof). INdeed peer review is not infallilble by anymeans. But perhaps the more correct statement at the moment is that there are no known problems with the proof. Remember Wiles's first proof had errors in it. Today Wiles's proof has, I beleive, been extended to other nonstable cases. Devlin wrote an artilce about this in his AMS collumn. He called them "left wing proofs" meaning 'sort of fuzzy and probably correct' as opposed to 'right wing proofs' whcih he categorized as short sharp and not to be argued with. other left wing proofs are the 4 colour theorem and the classification of finite simple groups. as far as we know they are to all intents and purposes correct and no one beleives that any error which may occur will be insurmountable.

no, that is false. in the UK legal driving age is 17 for cars (16 for motorbikes) and 18 for drinking and voting. ages of consent for sex are variable depending on the ages of both partners We do ahve armed police, you just aren't aware of them.

Rakista, do you understand the fallacy of cum hoc ergo propter hoc? Or more importantly, those who disagree with you are pointing out that you are implying a causal link that being of certain distinctions (such as of a race minority) causes criminal behaviour. That is not the case. Being teenage with all its rampant hormones and refusal to acknowledge restraint does predispose to criminal (or criminalized at any rate) or antisocial behaviour. Such reasoning as yours has been exactly used to justify pogroms and the very thing you claim to be against. and to create a silly example, it is well known that there is a link between low prices of bread and affordable housing, that doesn't mean that keeping bread cheap will make penthouses in manhattan affordable. I think most might agree that the curfews imposed upon people merely because of their ages are draconian and often unwarranted and do unfairly discriminate against people who have no ill intent. However, those are the rules that your society has imposed. True, that is not you direct decision but it is the decision of you parents and their generation and you do influence them since you can talk to your parents and make your case. If you wish for independence from them then yuo must accept that they have no responsibilty, financial or otherwise, for you. But, whilst you are their responsibility that is something you must accept. Perhaps when you are in a position to make a difference to those younger than you you will campaign to change the law. I know I for one do not think curfews on the young are good but i find your arguments indefensible. if you wished to mount a campaign of civil disobedience then why not remain out after curfew peacefully without breaking any other laws? that is the way to change opinions. Do not drink, nor smoke. Whilst i think the US laws on the legal age to drink are ridiculous that does not mean i agree you should break those laws by drinking in large groups of youths after a curfew. there are better ways to obtain acceptance. ghandi's biggest coup was showing that it was possible to take the moral highground and be successful. so, drink with meals in restaurants with parents, with your family, show that it can be acceptable to drink. and remember that adults are as liable to arrest fro drunk and disorderly behaviour too.

taylor series of what? (1-b^2)^{-1/2}? we aren't using the taylor series of that, we are using the series of (1-t)^{-1/2}. That is what i meant when i said it wasn't clear to what the "first order" referred. we are certainly using the first order estimate of the expansion of the latter. we haven't expanded the former directly.

we are using the first order expansion of (1+x)^{-1/2} but it so happens that we are replacing x with beta^2 so it is not linear in beta. i wouldn't like to say whether that makes it not first order since it isn't clear to what's order you are referring when you say "first order".

firstly hyperbolic trig and hyperbolic geometry are not necessarily related, indeed they aren't really. secondly neonblack is missing some h's in his post (sech is 1/cosh) but that;s just a typo. hyperbolic trig fucntions are as defined, they also satisfy integrals akin to theordinary trig ones, and can be related to ordinary trig by complex numbers. hyperbolic geometry: a geometry is a set of points with a rule for making geodesics. a geodesic is a "shortest" path. but perhaps you might wnat to think of it is a "path of least resistance" since you may well think that straight lines are always the "shortest" path. geometries satisfy some axioms, things like give two points there is a geodesic passing through them and so on. eulicdean geometry, the one we trhink in of straight lines and planes and such satisfies the parallel poostulate: given a geodesic and a poitn not on that geodesic there is a unique parallel geodesic through that point. for many years, centuries, it was beleived that the parallel postulate was deducible fromthe other axioms (of euclid) but it isn't. it turns out you can define other geometries satisfying all of euclid's rules except the parallel postulaate. indeed we live on one: spherical geometry. we know that planes fly not on straight lines but on great circles as teh shortest path. two great circles always inteserct. so the parallel postulate fails as there is no second geodesic. hyperbolic is the other extreme. there are several models of hyperbolic geometry. the easiest to me is the poincare disc: it is the inside of unit disc in the complex plane (all complex numbers of modulus less than one) and the geodesics are semicircles that intersect the boundary at right angles (note this includes diameters of the disc - cirlces of infinte radius). in this space, given a gedoesic and a point not on the geodesic there are an infinte number of "parallel" geodesics through that point. the easiest example is if the frst geodesic is not a diamter, and the point is the centre of the disc and then there are an infinite number of diameters through the centre not intersecting the geodesic.

what is "it"? no, how do you square things again? and why are you squaring something? no. 2sinh(x)cosh(x)=2(e^x-e^-x)(e^x+e^-x)/4 = (e^2x - e^ -2x)/2 = sinh(2x) that is how it goes. you can differentiate sinh and cosh right? look at their definitions sinhx=(e^x-e^-x)/2 you can differentiate that. given you can differentiate f(x) what is the derivative of f(3x)^2? nothing to do with hyperbolics, jsut your statndard calculus.

No. You were given the definition of S^n, the n-sphere. commonly we take radius 1 as the standard. With the inequality it is a ball (or sometimes it is called a disc).